The Eighth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories: Proceedings of the Meeting held at The Hebrew University of Jerusalem 22-27 June 1997 9810237936, 9789810237936

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THE EIGHTH

MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories

Also published by World Scientific:

PROCEEDINGS OF THE SIXTH

MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY PART A & PART B Eds. Humitaka Sato and Takashi Nakamura Series Ed. Remo Ruffini PROCEEDINGS OF THE SEVENTH

MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY PART A & PART B Eds. Robert T. Jantzen and G. Mac Keiser Series Ed. Remo Ruffini

PART A

THE EIGHTH

MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories

Proceedings of the Meeting held at The Hebrew University of Jerusalem 22-27 June 1997

Editor

Tsvi Piran The Racah Institute for Physics The Hebrew University of Jerusalem Jerusalem 91904 Israel

Series Editor

Remo Ruffini International Center for Relativistic Astrophysics University of Rome "La Sapienza" Rome 00185 Italy

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THE EIGHTH MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying. recording or any information storage and retrieval system now known or to be invented. without written permission from the Publisher.

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THE MARCEL GROSSMANN MEETINGS The Marcel Grossmann Meetings were conceived with the aim of reviewing recent developments in gravitation and general relativity, with major emphasis on mathematical foundations and physical predictions. Their main objective is to bring together scientists from diverse backgrounds in order to deepen our understanding of spacetime structure and review the status of experiments testing Einstein's theory of gravitation. Publications in the Series of Proceedings Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity - these volumes (Jerusalem, Israel, 1997) Edited by T. Piran World Scientific, 1998 Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity (Stanford, USA, 1994) Edited by RT. Jantzen and G.M. Keiser World Scientific, 1996 Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity (Kyoto, Japan, 1991) Edited by H. Sato and T. Nakamura World Scientific, 1992 Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity (Perth, Australia, 1988) Edited by D.G. Blair and M.J. Buckingham World Scientific, 1989 Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity (Rome, Italy, 1985) Edited by R Ruffini World Scientific, 1986 Proceedings of the Third Marcel Grossmann Meeting on General Relativity (Shanghai, People's Republic of China, 1982) Edited by Hu Ning Science Press - Beijing and North-Holland Publishing Company, 1983 Proceedings of the Second Marcel Grossmann Meeting on General Relativity (Trieste, Italy, 1979) Edited by R Ruffini North-Holland Publishing Company, 1982 Proceedings of the First Marcel Grossmann Meeting on General Relativity (Trieste, Italy, 1976) Edited by R Ruffini North-Holland Publishing Company, 1977 Series Editor: REMO RUFFINI

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Yuval Ne'eman

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SPONSORS The Hebrew University of Jerusalem International Center for Relativistic Astrophysics (ICRA) International Union of Pure and Applied Physics (IUPAP) The Israeli Ministry of Science The Italian Ministry of Foreign Affairs National Science Foundation - USA (NSF) The Technion Tel Aviv University UNESCO

ORGANIZING BODIES OF THE EIGHTH MARCEL GROSSMANN MEETING:

INTERNATIONAL ORGANIZING COMMITTEE David Blair, Yvonne Choquet-Bruhat, Dimitris Christodoulou, Thibault Damour, Jurgen Ehlers, Francis Everitt, Fang Li Zhi, Stephen Hawking, Yuval Ne'eman, Remo Ruffini (Chair), Humitaka Sato, Rashid Sunayev, Steven Weinberg

LOCAL ORGANIZING COMMITTEE T. Piran (chair), J. Bekenstein, M. Carmeli, A. Casher, A. Dar, A. Dekel, J. Horwitz, J. Katz, M. Milgron, S. Nussinov, A. Ori, C. Sonnenschein

LOCAL SCIENTIFIC SECRETARIAT S. Ayal, E. Cohen, Y. Friedman, J. Granot, H. El-Ad, S. Hod, S. Kobayashi, R. Sari

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INTERNATIONAL COORDINATING COMMITTEE for coordinating national travel funding application, soliciting program ideas, and facilitating information dissemination Bob Jantzen (chair) Argentina: Chimento L.P., Jakubi A.S.; Armenia: Gurzadyan V.; Australia: Fackerell E.D., Lun T., Manchester R., Opat G., Scott S., Szekeres P., Austria: Urbantke H.; Azerbaijan: Seidov Z.F.; Belgium: Henneaux M.; Belorussia: Inkevich A.V.; Bolivia: Aguirre C.; Brazil: Aguiar 0., Novello M., Villela T.; Canada: Cooperstock F., Israel W., Unruh W.; Chile: Teiteiboim C.; China: Gao J.G., Cheng Fu Zhen, Li Qibin; Colombia: Torres S.; Costa Rica: de Teramond G.; Croatia: 1¥tdic Z.; Czeck Republic: Bicak J.; Denmark: Novikov I.; Ecuador: Hoeneissen B.; EI Salvador: Violini G.; Estonia: Einasto J.; France: Deruelle N., Iliopoulos J., Mignard F.; Georgia: Kharadze E.K.; Germany: Hehl F.H., Hillebrandt W., Neugebauer G.; Greece: Kotsakis, S.; Hong Kong: Yu A.; Hungary: Perjes Z.; India: Chitre S.M., Narlikar J., Radakrishnan, V.; Ireland: O'Murchada N.; Israel: Finzi A., Jamer M., Peres A.; Italy: Preparata G., Regge T., Treves A.; Japan: Fujimoto M., Futamase T., Nakamura T., Sasaki M., Sato K., Tomimatsu A.; Kazachstan: Abdildin A.M.; Korea: Cho Y.M., Lee C.H., Song D.J.; Kyrgystan: Gurovich, V.Ts.; Lithuania: Pyragas K.A.; Mexico: Plebanski J., Rosenbaum, M., Ryan M., Moldova: Chernigovsky S., Gaina, A.; Morroco: Chamcham K.; Netherlands: Berends F.A.; Poland: Demianski M., Sokolowski L., Trautman A.; Romania: Visinescu M.; Russia: BisnovatyiKogan G.S., Boyarchuck A.A., Chechetkin V.M., Khriplovich Yu.B., Melnikov V., Starobinsky A.A.; Slovenia: Cadez A.; Spain: Ibanez J., Perez-Mercader J., Verdaguer E.; South Africa: Maharaj S.; Sweden: Rosquist K.; Switzerland: Hajicek P., Jetzer P.; Taiwan: Ni W.T.; Turkey: Nutku Y.; UK: Barrow J., Gibbons G.W, Ross G.; Ukraine: Fomin P.I.; USA: Ashtekar A., Bardeen J., DeWitt-Morette C., Finkelstein D., Frieman J., Halpern L., Hellings R., Klauder J., Mashhoon B., Nordtvedt K., Parker L., Schwarz J., Shapiro I., Smoot G., Teukolsky S., Thorne K.S., York J.; Uzbekistan: Zalaletdinov R.M.; Vatican City: Stoeger W.; Venezuela: Percoco U.; Vietnam: van Hieu Nguyen; R. Yugoslavia: Sijacki D.

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MARCEL GROSSMANN AWARDS

EIGHTH MARCEL GROSSMANN MEETING

Institutional Award

THE HEBREW UNIVERSITY OF JERUSALEM ''for its role as a cradle of Science and Humanities and for hosting the manuscripts of Albert Einstein"

Individual Awards

TULLIO REGGE ''for his contributions to the interface between mathematics and physics leading to new fields of research of paramount importance in relativisic astrophysics and particle physics"

FRANCIS EVERITT ''for leading the development of extremely precise space experiments utilizing superconducting technology to test General Relativity and the Equivalence Principle"

Each recipient is presented with a silver casting of the TEST sculpture by the artist A. Pierelli. The original casting was presented to His Holiness Pope John Paul II on the first occasion of the Marcel Grossmann Awards.

x FOURTH MARCEL GROSSMANN MEETING Institutional Award THE VATICAN OBSERVATORY Individual Awards WILLIAM FAIRBANK ABDUSSALAM

FIFTH MARCEL GROSSMANN MEETING Institutional Award THE UNIVERSITY OF WESTERN AUSTRALIA Individual Awards SATIO HAYAKAWA JOHN ARCHIBALD WHEELER

SIXTH MARCEL GROSSMANN MEETING Institutional Award RESEARCH INSTITUTE FOR THEORETICAL PHYSICS (Hiroshima) Individual Awards MINORU ODA STEPHEN HAWKING

SEVENTH MARCEL GROSSMANN MEETING Institutional Award THE HUBBLE SPACE TELESCOPE INSTITUTE Individual Awards SUBRAHMANYAN CHANDRASEKHAR JIM WILSON

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The recipients of the Marcel Grossman Awards (from top anticlockwise): Menachem Magidor, Francis Everitt, Thllio Regge.

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Photograph of the TEST sculpture of A. Pierelli by S. Takahashi.

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PREFACE The Eighth Marcel Grossmann Meeting (MG8), dedicated to Yuval Ne'eman on the occasion of his 70th birthday, took place June 22-27, 1997 at the Givat Ram campus of the Hebrew University of Jerusalem, Jerusalem, Israel, where Einstein's archives are kept today at the National Library. Albert Einstein, who helped found the university, was a member of the first board of trustees. The opening ceremonies were held on the evening of June 22nd with welcoming addresses by Remo Ruffini (Chairman of the International Organizing Committee), Menachem Magidor (President of the Hebrew University), David Harari (representative of UNESCO) and Humitaka Sato (representative of IUPAP). The three Marcel Grossmann Awards were presented by Remo Ruffini and Humitaka Sato. The Hebrew University of Jerusalem was the recipient of the institutional award, which was accepted on behalf of the Institute by Menachem Magidor. The individual award recipients were Francis Everitt and Tullio Regge. Each received a silver replica of the TEST (Traction of Events in Space-Time) sculpture by Attilio Pierelli. The audience received an elegantly printed award pamphlet including pictures of the individual recipients and some spectacular glossy photographs of the TEST sculpture taken by the artist Shu Takahashi, who helped design the two meeting posters and the meeting tote bag. Also included in the pamphlet was an essay by Anna Imponente about the interaction between science and art which is concretely embodied in the TEST sculpture, the three-dimensional extension of the Marcel Grossmann Meeting logo used in the meeting posters and other promotional materials since MG4. After the opening ceremonies, the meeting began with a public talk by Yuval Ne'eman entitled "On the clouds in the sky of Physics at the end of the twentieth century." The scientific program included 25 morning plenary talks during five days and 40 parallel sessions over four afternoons, during which approximately 400 papers were presented. No poster sessions were held. Instead, everyone not able to be allotted a longer time to speak was guaranteed their "at least five minutes of fame" to announce their work for further discussion on an individual basis. Nearly six hundred participants and accompanying persons were present for the meeting during a week of truly fine weather in Jerusalem. The participants were able to enjoy not only the scientific discussion but also to visit the three thousand year old city of Jerusalem. During the afternoon of June 25th the participants enjoyed a tour that passed through the new and the old sections of the city, visiting some of the holiest sites for Christianity, Judaism and Islam. On this excursion could be seen the location that was considered for generations as the "center of the Universe," a concept of particular interest to relativists. The combined efforts of the local organizing committee went into making the meeting become a reality. The assistance of the enthusiastic local scientific secretariat, S. Ayal, E. Cohen, Y. Friedman, J Granot, H. EI-Ad, S. Hod, S. Kobayashi, and R. Sari in every single aspect of the meeting, from preparation of the world wide web pages to the electronic registration procedure and the publishing of the proceedings, was essential to the success of this conference. The International Coordinating Committee's one hundred members from nearly 60 countries (chaired

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by Bob Jantzen) also offered significant input to the scientific program during the planning stages and also aided participants from their nations with national travel funding application and visa information. The farewell banquet was held in the outdoor sculpture garden of the Israeli Museum, overlooking the Givat Ram campus of the Hebrew University and the mountains of Jerusalem. Those present enjoyed an Middle Eastern style dinner and were treated to a wonderful after-dinner talk by Ivor Robinson. On June 27th Remo Ruffini closed the meeting, using the occasion to announce the establishment of a UNESCO supported network of International Centers for Relativistic Astrophysics: the ICRA Network. A warm expression of thanks was given to the participants, speakers, and organizers, and to all those whose financial help made possible the realization of this Eighth Meeting, especially the American NSF (and to David Brown for applying for the NSF funds), the Italian Ministry of Foreign Affairs, IUPAP, UNESCO, the Israeli Ministry of Science and the Israeli Academic Institutions which together enabled a significant number of participants to attend. Tsvi Piran

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DEDICATION IN HONOR OF YUVAL NE'EMAN'S 70TH BIRTHDAY On behalf of the International Organizing Committee (IOC), it gives me great pleasure to dedicate to Yuval Ne'eman this Eighth Marcel Grossmann Meeting (MG8) held in Jerusalem June 22-27, 1997. Since MG2 in 1979, Yuval has been a very active member of our IOC. Through these years it has been wonderful to collaborate with him on the many aspects of the definition of the scientific programs as well as in overcoming problems and, at times, obstacles in the international organization of these meetings. Most memorable in this respect was the planning and organisation of MG3 in Shanghai: nothing less then the new concept of a "scientific passport", signed by Abdus Salam and recognized by the Chinese Foreign Ministry, had to be invented in order to allow Israeli scientists to participate in MG3. I still remember the solution of that problem, achieved in close contact and collaboration with Chuo Pei Yuan in China and Yuval in Israel, as the most difficult negotiation in my career. On many occasions, in Israel and abroad, during the MG meetings and in informal circumstances, reading the scientific literature and in discussions with colleagues, I became acquainted of the most remarkable achievements and extraordinary episodes of the life of Yuval Ne'eman. Nevertheless I could not have written this remembrance without his help which, also on this occasion, he has generously offered. Background

Yuval Ne'eman was born in 1925 in Tel-Aviv, Israel (then "Palestine", a League of Nations British Mandate), a scion of a family which had returned (from Lithuania) to the old Jewish homeland in the beginning of the XIXth century. His grandfather, Abba Ne'eman, a self-educated engineer, was one of the 66 founders of the city of Tel-Aviv (1909) and established in 1900 a pumps factory (for water supply, urban and agricultural) one of the first industrial workshops in the country (it closed in 1987). Yuval's father,· Guedaliahu, also an engineer, worked in the factory and directed it after Abba fell ill in 1939. Thus, when Yuval matriculated in 1940 from Herzliah Highschool in Tel-Aviv (the first Hebrew highschool in modern Israel) at the age of 15-the youngest ever matriculating at this school-it was almost selfevident that he was going to study mechanical and electrical engineering at the Haifa Technion (now the Israel Institute of Technology) in order to enter the family factory and eventually take over from his father. The Technion had a minimal admission age of 16, so Yuval spent the year 1940-41 working in the family factory, half the time in the machine shop and the other half helping design new machinery. The Second World War had started, Palestine and the British Middle East were cut off from Britain, and the country's factories were requested to help in the war effort. The Ne'emans were now producing petrol pumps for gas stations, for aircraft refueling, etc. It was also during that year that Yuval enlisted in the Haganah, the Jewish (underground) defense organization, which later, at the creation of the state, became the nucleus of Israel's army. The third development during that year bears

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a closer causal relation with our topic. At high school, Yuval had been attracted by mathematics and had even done some original work-the mechanics of the next arithmetical operation (after exponentiation and its two inverses, the extraction of the n-th root and of the logarithm). However, he had not been attracted to physics until in 1940, S. Samboursky of Hebrew University (Jerusalem) gave a series of popular lectures on Modern Physics. This caused Yuval to read Eddington's "Nature of the Physical World" (it had been translated into Hebrew) and to fall in love with Modern Physics [note that S. Samboursky arrived in Israel in 1924 after taking his Ph.D. in Germany, under the guidance of T. Kaluza). By that time Yuval had also acquired an encyclopedic knowledge of History, Geography, Linguistics, etc. In the beginning of each school year, his parents would buy the relevant textbooks. Yuval would gobble them up in the first fortnight, solve the problems and have nothing much to learn in class during the rest of the year. Teachers who understood that he was bored in class encouraged him to spend many hours at the Municipal Library and Ahad-ha'am Archives which were contiguous with the school. Moreover, for his "bar-mitzvah" (Jewish "coming of age" at one's 13th birthday) Yuval received from his parents a Britannica, in which he became permanently immersed for the next three years. Throughout 1941-1945 Yuval combined studies of Mechanical and Electrical Engineering at the Haifa Technion with active service in the Haganah, first training to fight a guerrilla warfare against the expected German occupation (Field-Marshal Rommel was advancing from Lybia into Western Egypt), then becoming a smallarms instructor-eventually (1943) training, among others, Hannah Senesh, who was parachuted into Hungary, in a mission combining the establishing of a liaison with some allied agents with, on the other hand, reporting what was happening to Hungary's Jews (half a million died in the gas chambers in 1944) [Senesh was caught by the Germans and executed.) Yuval graduated (B.Sc. Eng.) in 1945 and took a "Diploma of Engineering" (German style, a more advanced degree) in 1946. In between, he also graduated from the Haganah's Officer's School. The Military Period

By that time, Yuval had decided to go into Physics, except that he felt he also had a more urgent duty, namely the struggle for Jewish immigration and settlement and for the future Jewish state. To understand the background, we provide the main facts-as perceived from Yuval's angle. During World War I, late in 1917, looking for Jewish political support in the USA, Britain had issued the Balfour Declaration, promising the establishment of a Jewish Homeland in Palestine, then under Turkish rule. The Declaration was embodied in the League of Nations Mandate in 1921 and hopes were high, immigration increased and the arriving pioneers settled land as it was purchased by the Jewish National Fund. The Arab population in Palestine, however, objected to the entire idea. Led by the Grand Mufti of Jerusalem, they almost immediately launched an armed struggle, attacking the Jewish population everywhere in the country. In 1921, 1929 and again in 1936-39, the local Arabs, aided by volunteer units from neighboring Arab countries, conducted an organized offensive against the "Zionist bridgehead". The Haganah was the Jewish military

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answer, a poorly armed militia, which managed to protect the Jewish population with British blessing until 1938. In that year, Britain reversed its policies and issued a "White Paper" forbidding Jewish immigration, purchase of land and settlement. The Haganah and two splinter organizations turned against the British. Jews in central Europe were being persecuted and were soon going to be annihilated by Nazi Germany. Many thousands attempted to flee and go to Palestine, except that the Royal Navy was now patrolling the coasts, catching refugee ships and returning them to Germany. When World War II started, however, the Haganah postponed action against the British: Germany should be beaten first. After 1946, the British were again the adversary, the new UK Labor cabinet having renewed its commitment to that "White Paper." Returning now to Yuval, in his readings in History he had noted that in periods of crisis, eminent scientists had managed to combine a career in science with one in the armed forces. Fourier was a general and so were Lazare Carnot (creator of Ecole Poly technique and Ecole Normale and father of the thermodynamicist) and Benjamin Thomson (Lord Rumford). Benjamin Franklin and Lavoisier partook in the political leadership of their respective countries. These examples served to preserve the hope of returning one day to studies in Physics and kept him on the lookout for the right timing. The year 1946-47 was spent partly in the Pumps Factory-where Yuval developed three new models, one of which can still be found working in the field-and in military activities, first as a platoon, then a company commander, protecting either the landing of refugee ships or the erection of new settlements. By mid 1947 he was fully mobilized. On 29 November 1947, the UN General Assembly decreed the partition of Palestine and the establishment of a Jewish and an Arab state. England announced it would evacuate Palestine on May 15, 1948-and Yuval was already preparing to renew his studies-when war started, as the Arabs rejected partition and attacked the Jewish entity everywhere. The War of Independence lasted 15 months and was Israel's bloodiest and hardest, with 6,000 dead (l entire Jewish population) and with several moments when the Jewish side appeared close to a final defeat, especially when in May 1948, Egypt, Syria, Jordan, Iraq and Lebanon invaded in force, helped by contingents from all over the Arab World. Yuval fought, first at the infantry battalion level, leading his battalion in two well planned and successful battles, then as chief of operations of a brigade. In the latter capacity, he managed to develop novel tactics, adapted to the status of Israel's armament at that time, with almost no artillery and no armor. After the war he was moved to the High Command, first as Chief of the Operations section, then as Director of Planning (1952-55). In a lecture at the MG6 Conference, he told us something of his experiences, in that capacity, in "designing a country" -deciding where to site the new cities, villages, forests etc. On the strategic side, he crystallized the basic doctrine which was followed by Israel till after the 1967 Six-Day War. That war itself was fought according to the contingency plans he had prepared in 195354. In 1955 he was appointed Deputy, then Acting, Director-General of Military Intelligence.

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Back to Science-in London

The year 1956 brought the Sinai-Suez war-Yuval was heavily involved in Israel's connection with France in that conflict. The 1956 victory brought about a more relaxed atmosphere in defense thinking, and Yuval now felt this was his opportunity, perhaps his last chance, of becoming a scientist. He had taught himself General Relativity and was captivated by the beauty of the geometrical content, of the mathematical realization of the Principles of Covariance and of Equivalencethough he had as yet no knowledge of Group Theory as such. Nathan Rosen (of the "Einstein-Rosen bridge", the Einstein-Rosen cylindrical gravitational wave and the EPR "paradox") had immigrated in 1954 and had now founded a Department of Physics at the Technion, and Yuval asked General Moshe Dayan, the Israeli Chief of Staff, for a two-year leave, in order to return to the Technion and work under Rosen in General Relativity. Dayan asked "could you do it in England?" The position of Defense Attache at the Israeli Embassy in London was unoccupied and the very unconventional Dayan suggested that Ne'eman combine the Attache's duties with studies in London. Yuval agreed. Once in London, however, he did some reconnoitering and realized that with the Embassy being in Kensington, it would be impossible to work at Kings, east of Trafalgar square, where Bondi taught and where the General Relativity activity was centered. Instead, he found Imperial College within five minutes walking distance from the Embassy. Under "Theoretical Physics", the catalogue listed a Prof. Blackman (a well known Solid State physicist). When Ne'eman asked him whether anybody was working on "Unified Field Theory" (Einstein's last endeavor), Blackman said "I don't know about 'Unified' field Theory, but Abdus Salam in Mathematics is working on 'Field Theory'." This was how Yuvallanded in Salam's group in January 1958. Throughout the spring of 1958 Yuval managed to attend about one half of the lectures, studying the rest from notes taken by R. Streater, a young graduate student. July 1959 was eventful in the Middle East and the job at the Embassy became very demanding. This was not what Dayan had promised and Yuval complained. It was decided that he would be freed from the Attache's position altogether and have a year of pure studies, at the Israel AEC's expense. His replacement indeed arrived and in May 1960 Yuval was at last free to concentrate his efforts on Physics alone. By October he had achieved his first major result. Understanding the pattern: SU(3) and the Octet Model

Particle physics was born with the discovery of the neutron in 1932. By 1935, this had lead to the realization that new, short ranged interactions were involved. Yukawa and Fermi extracted the dynamical mechanisms, effective models for two such, interactions, but by 1955 the list of new particles included some 30 species and after the construction of improved bubble chambers (1959) it increased at an accelerated rate, reaching 3 digit figures. Mass, Spin, C and P parities, together with Isospin and Strangeness, served to characterize any species, but the variety was bewildering. Against Salam's advice ("You are embarking on a highly speculative venture and your one year fellowship may be over with nothing to show!"), Yuval set out on a

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search for an understanding of the pattern. Taking an as yet untried algebraic route, he mastered the theory of Lie algebras and studied Cartan's 1894 classification of the simple ones, noting that what he was after was an algebra of rank r = 2 to acc(')mmodate Isospin and Strangeness. By October 1960 he had identified SU(3) as the classifying symmetry of the hadrons, the particles experiencing the Strong Interactions, with the nucleons sitting in an octet representation-rather than in the defining triplet. Mesons also appeared in octets; the excited baryons (higher in mass and spin) filled additional octets-and multiplets with either 10 or 27 states. Aside from the classification, the symmetry provided dynamical information-coupling strengths, analogous to the role of the electric charge in Coulomb's or Lorentz's flaws in Electromagnetism. The symmetry is also embodied in an eightfold set of conserved charges, with approximately universal couplings to eight (massive and short-ranged) potentials [1]. The discovery of the SU(3) symmetry and classification has often been compared to Mendeleyev's 1868 classification of the chemical elements. In identifying his pattern, Mendeleyev had been forced to leave a number of unfilled squares in his "Periodic Chart," and it was the subsequent discovery of chemical elements, fitting precisely the characteristics of these empty squares as required by the classification, which provided the Chart's validation. In the same manner, Yuval's SU(3) multiplets contained several unassigned states, and it was the fulfillment of these predictions-and especially the discovery, early in 1964, of the n- hyperon, with the predicted [2] mass (1675 MeV) spin (3/2), isospin (0) and strangeness (-3), which provided the proof for SU(3) [3].

Understanding the Structure: Quarks The Periodic Chart of the Chemical Elements was explained when the structure of the atom was understood in terms of its charged constituents (protons and electrons) in the years 1906-1925 by Rutherford, Bohr and Pauli. Similarly, Unitary Symmetry (SU(3) in the baryon octet version) was explained in 1962-73, in terms of constituents. In their recent biography of Feynman [4], John and Mary Gribbin write: "The first tentative steps towards the idea of a deeper layer of particles within the hadrons was made in 1962 by Ne'eman (then working for the Israel Atomic Energy Commission) and his colleague Haim Goldberg-Ophir. They wrote a paper suggesting that baryons might each be made up of three more fundamental particles, and sent it to the journal Nuovo Cimento, where it was mislaid for a time, but was eventually published in January 1963 (though the IAEC preprint was circulated internationally in March 1962). The paper attracted little attention, partly because the Eightfold Way itself had not yet been fully accepted, but also, as Ne'eman has acknowledged, 'because it did not go far enough'. The authors had not yet decided whether to regard the fundamental components as proper particles or as abstract fields that did not materialize as particles." In "Quarks for Pedestrians" [5], H.J. Lipkin has a slightly different evaluation. He writes: "Goldberg and Ne'eman then pointed out that the octet model was consistent with a composite model constructed from a basic triplet, with the same

xx isospin and strangeness quantum numbers as the sakaton (a model suggested by S. Sakata, according to which all hadrons are made of protons, neutrons and lambda hyperons, R.R.), but with baryon number 1/3. However, their equations show that particles having third-integral baryon number must also have third integral electric charge and hypercharge. At that time, the Eightfold Way was considered to be rather far-fetched and probably wrong. Any suggestion that unitary symmetry was based on the existence of particles with third-integral quantum numbers would not have been considered seriously. Thus, the Goldberg-Ne'eman paper presented this triplet as a mathematical device for construction of the representations in which the particles were classified. Several years later, new experimental data forced everyone to take SU(3) more seriously. The second baryon multiplet was found, including the n- ... Gell-Mann and Zweig then proposed the possible existence of the fundamental triplet as a serious possibility and Gell-Mann gave it the name of quarks." The Gribbins write: "Zweig regarded these, from the outset, as real particles, not 'abstract fields'. Gell-Mann was much more cautious, and trod a path almost exactly halfway between the confident espousal of aces (quarks, R.R.) as real by Zweig, and the dismissal of the 'fundamental components' as 'abstract fields' by Ne'eman and Goldberg-Ophir. Like Zweig he gave ... a name (quarks), but like the Israeli team he expressed reservations about their reality." M. Gell-Mann had in fact arrived in 1960-61 independently at the same SU(3) octet classification as Ne'eman, though somewhat later. He submitted his paper for publication in The Physical Review in March 1961 (Yuval's was submitted to Nuclear Physics in February 1961) In June 1961, at a conference at La Jolla, attended by Salam and by Gell-Mann, some experimental evidence appeared to contradict the octet assignment-and Gell-Mann withdrew his paper altogether. When, told by Salam of these news, Ne'eman chose to stand by his model. Gell-Mann later wrote a new paper, in which he presented the octet and the Sakata models as two open possibilities (submitted in September 1961 [6]). At the CERN conference in July 1962, both Ne'eman [2] and Gell-Mann [7] predicted the properties of the nand suggested the experiment. Gell-Mann's original SU(3) preprint was finally published in 1964 in a collection of reprints issued by Gell-Mann and Ne'eman under the (same) title The Eightfold Way [8]. Between 1961 and 1964 the experimental data overwhelmingly validated the octet model. Summarizing their discussion of Unitary Symmetry, the Gribbins write: "For this and his other work on the classification of fundamental particles, Gell-Mann received the" 1969 Nobel Prize for Physics; surprisingly, the Nobel Committee overlooked Ne'eman." There are now some twenty texts describing the rise of the Standard Model, the 1975 grand synthesis of the particle interactions, excluding gravity; they all open with the discovery of SU(3) and the conception of the quarks.

The Symmetry Breaking: the "Fifth Interaction" (now "Higgs Sector") The dynamical methodology appropriate to Particle Physics is Quantum Field Theory, perfected in 1946-48, when it was applied to Quantum Electrodynamics ("QED"). Between 1955 and 1971, however, Quantum Field Theory came into

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disrepute. On the one hand, it appeared unutilizable in the Strong Interactions, since there was no way of using a perturbative expansion, with a coupling such as the nucleon-pion coupling, with a value of 14 On the other hand, early attempts to use the Yang-Mills model (a local gauge theory) indicated that Quantum Field Theory was not unitary, off-mass-shell. Although the latter problem was resolved by Feynman in 1962 through his introduction of "ghost" fields, most workers in the field at that time chose not to use QFT when dealing with Strong Interactions and applied the on-mass-shell dynamics of S-matrix Theory instead. In an S-matrix formulation, however it seemed impossible to have a broken symmetry of the Strong Interactions. There was no problem with the successes of SU(3) in predicting magnetic moments or weak decays, because these were due to weak-coupling interactions, obeying QFT and thus allowing for a perturbative approach (the formulae were first-order results). The similarly successful mass formulae and intensity rules obtained for the broken SU(3) Strong Interactions were puzzling, however. How could first-order perturbative results hold in a strong coupling theory, that of the Strong Interactions? The riddle was resolved by Ne'eman in a 1964 paper [9], entitled The Fifth Interaction. The suggestion was that the Strong Interaction itself is invariant under SU(3), whereas the SU(3)-breaking indeed originates in a different ("fifth") interaction, with a weak coupling. Ne'eman's approach was incorporated in 1976 within the "Standard Model", in which QCD, the fundamental Strong Interaction, is flavour-invariant, namely [SU(3)colour, SU(3)/lavour] = 0 and it is assumed that the SU(3)-breaking is induced by the quark masses. These mass differences are an input and the simplest assumption is that they originate in the "Higgs sector" , namely one scalar field and one Yukawa coupling per (quark or lepton) mass. This is then the ad hoc present identification of Ne'eman's Fifth Interaction. Kaluza-Klein Geometrization and High Energy Algebraics of the Quark Model The first direct successes of the Quark Model (beyond SU(3) symmetry) were derived by Guersey and Radicati in 1964, based on a Wigner-like supermultiplet approach, introducing SU(6) as a tensor-product of SU(3) with nonrelativistic SU(2)spin, a space-time symmetry. Earlier, Ne'eman had suggested fusing internal and external symmetries, using an extension of the technique used by Kaluza and Klein in the twenties to reinterpret electric charge as a fifth dimension. A seminar he organized in Dallas in the spring of 1954 [10] dealt with the embedding problem, both local and global. With the success of the static quark model, however, this approach was discontinued. It was revived in 1971, when the NeveuSchwarz-Ramond superstring was shown to require 10 dimensions, precisely the minimal number imposed by local embedding considerations. The Kaluza-Klein approach was further developed in the eighties for supergravity and is now the backbone of "M-theory". Returning to the Quark Model, Ne'eman, together with N. Cabibbo and L.p. Horwitz, discovered a series of predictions relating to the highenergy "asymptotic" region [11]. Several groups had probed that region-Levin and Frankfurt, Lipkin and Scheck, Kokiceddee and Van Hove-but the "CHN" (Cabibbo-Horwitz-Ne'eman) results uncovered the algebraic structure, with univer-

xxii

sality features, for which Ne'eman later provided dynamical derivations [12], which were finally explained in terms of QCD. An extension of this work ("CHKN" [13]) further explored the analytical structure, raising the possibility of the Regge intercept for the Pomeranchuk trajectory is Re a(O) = 1 - E, i.e somewhat lower than the Froissart bound. This idea became the foundation of a perturbative expansion in "Regge field theory" in terms of the small parameter L Spectrum-Generated Algebras, Quadrupoles and Regge Sequences

With Y. Dothan and Gell-Mann, Ne'eman introduced in 1965 [14] the method of spectrum-generating algebras, an extension of the idea of symmetry, namely the identification of the entire spectrum of solutions of a quantum problem with one representation of a group (noncompact for infinite spectra) and the construction of the algebraic operators in terms of the variables of the specific problem. The method provided a new understanding of well-known problems in Quantum Mechanics, but its most spectacular application was in the construction of the superstring [15]. Applications span most physical sectors (condensed matter, molecular, atomic, nuclear, particles [16]). Another important application was the identification of the hadron's sequences of excitations in angular momentum (Regge sequences) with unitary infinite-dimensional representations of SL(3,R). The identification in 1963 was limited to bosonic sequences, whether nuclear or hadronic, but in 1969, together with D.W. Joseph, Ne'eman constructed representations of the double covering, i.e., spinors [17]. There was still some mystery with respect to the allowed sequences, mainly whether a sequence starting with J = 3/2 exists-it was later shown to be singular. The full classification of the unitary representations of SL(3, R) (including those of the double-covering) was supplied by Dj. Sijacki in 1975 (18). The structure of the algebraic operators was shown to involve time-derivatives of quadrupolar excitations-an interpretation which was successfully validated in nuclei by Biedenharn and collaborators [19]. Many years later (1990), Ne'eman (together with Sijacki) provided a mathematical derivation of the algebra for the hadron case, based on QCD [20-23]. This is the algebra of Diff( 4,R), represented as in gravity, over its sl (4,R) subalgebra and contained as a subalgebra in the (infinite group of) the su(3) colour gauge, in the IR limit. Ne'eman has named it Chromogravity. It thus generates an intrinsic "gravity", which could explain both the Regge trajectories' quadrupolar excitations and colour confinement. The idea has been recently developed by other groups (D.Z. Freedman, K. Johnson, D. Singleton, F.A. Lunev, etc.). Discovery of Linear, Affine and World Spinors; Metric-Affine Gravity and Quantization

This interest in SL(3,R) led Ne'eman several years later to the discovery of curvedspace spinors. For some fifty years, it was believed that there could be no construction of spinors over curved space, and that Lorentz group spinors constructed over local tangent frames are the only possibility. Most textbooks in General Relativity published in those years [24] contain a sentence stating that "the linear group

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SL(n,R) has no double-covering," sometimes with a reference to E. Cartan. Studying affine generalization of gravity, with the Lorentz group replaced by the linear group as the symmetry on the frames [25], Ne'eman realized that the supposed restriction is erroneous (witness the 1969-75 construction of the representations of the double-covering of SL(3, R)). Ne'eman then proved the existence of a double covering for both SL(n,R), SA(n,R) and Diff(n,R) [26-28], with infinite-dimensional unitary spinorial representations, including world spinors. In the more recent texts by p. Budinich and A. Trautman [29] and D. Finkelstein [30], the new possibility is discussed. Budinich-Trautman write: " ... This road to spinors may be called topological: it is related in an essential way to the nontriviality of the fundamental groups 1fl of the groups of rotations. It has the virtue of allowing a generalization of the notion of spinorial representations to general linear groups (Ne'eman 1978) ... The group GL +(n,R) for n > 2 has no finite-dimensional faithful representations. In other words, spinors associated with the general linear group have an infinity of components. They have the virtue of not requiring, for their definition, any quadratic form or scalar product. They can be contemplated on a "bare" differentiable manifold without metric tensor. The topological approach to spinors is more general than the one based on the idea of linearization of a quadratic form." And Finkelstein comments: "The dramatic way to general-relativize spinors is to add extra components until we get a direct-sum representation that can be extended. After all, to special-relativize electric fields, we combined them with magnetic fields to make 6-vectors; and to special-relativize the Pauli 2-spinors, we combined them with anti-spinors to make Dirac 4-spinors. But in the case at hand, it takes an infinite-dimensional representation of GL 4 to be double-valued. We would need infinitely many physical partners for each spinor particle. A few brave people presently explore this domain, especially Y. Ne'eman ... " With A. Cant and Sijacki, Ne'eman has also constructed field representations ("Manifields") using deunitarized versions of the above representations [31-33]. One application has consisted in the further development of Metric-Affine Gravity (MAG), with F.W. Hehl and collaborators [34-36]. Basically linear, affine or world spinors are applicable to three types of physical problems: (l)providing matter fields in non-Riemannian theories, as in the MAG above; (2) describing hadrons in Einsteinian gravity at the phenomenological level (protons after all are not Dirac spinors-witness their magnetic moments and strong gravitational fields will excite their Regge recurrences); (3) the study of QCD in the IR region, i.e, Chromogravity. Having developed a nonRiemannian alternative model, Ne'eman used it to advance the Quantum Gravity program. Gravitational lagrangians are known to be finite, but suffer from a breakdown of unitarity, caused by p-4 propagators. The latter result from the Riemannian condition Dg = 0, relating the metric 9 to the connection r. By assuming that at high-energy (short-distance) gravity is nonRiemannian and obtaining the Einstein theory as an effective low-energy theory through spontaneous symmetry breakdown [37], Yuval was able (with C.Y. Lee) to prove finiteness of the overall theory. The program is as yet unfinished, since unitarity has not been proven yet, but Yuval's innovative contribution in these proceedings [38] has now brought in the new methodology of the super connection and noncommutative geometry. Let us trace these developments.

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In 1979, Yuval (and independently D. Fairlie) [39) conceived of the application of the supergroup SU(2/1) as a consolidated description of the electroweak SU(2)xU(I), including the spontaneous symmetry breakdown. The superconnection is valued over the defining SU(2/1) supermatrices, with the one form gauge potentials valued over the even parts of the supergroup and the Higgs zero-forms valued over the odd parts (so that the overall grading is odd everywhere). The unexplained mystery consisted in the lack of Bose/Fermi transitions in the matter representations (otherwise perfect, including the prediction of three lepton versus four quark states within one generation), the group grading following the chiralities instead. The system was also generalized so as to include SU(3)colour and the generations. With J. Thierry-Mieg, Ne'eman developed the mathematical construction [40). The same structure was introduced independently a few years later in mathematics by Quillen [41) and then rederived by Coquereaux, Scheck and others [42) within the framework of Connes' noncommutative geometry, by modifying the input in Connes' own rederivation of the electroweak theory [43). In [38) a superconnection "gauges" the hyperexceptional simple supergroup P(4,R), with even subgroup SL( 4,R) and appropriate Higgs fields leaving only SL(2,C) invariant. Supersymmetry, Supergravity and Superconnections

Space-time supersymmetry was introduced by Golfand and Lichtman in 1971 and by Wess and Zumino in 1973. The mathematical foundations, however, were very unclear. Ne'eman was immediately interested and with the mathematicians L. Corwin and S. Sternberg [44) provided the necessary algebraic basis. As a direct result, V. Kac was able to classify the simple superalgebras [45), including one exceptional family, the Q(n) discovered by Ne'eman and appearing as an example in the CNS paper [44). One eventual continuation of this program was the discovery of Super-gravity by two groups [46). Ne'eman and Gell-Mann had set out on the same route, but had taken an algebraical rather than a geometrical approach. As a result of their work, they concluded that the N = 8 version belonged to a type (N = 4jmax) which was very constrained algebraically and had good chances of being finite. The N = 2, J max = 1/2 Wess- Zumino model has indeed one less renormalization constant, the N = 4, jmax = 1 super symmetric Yang-Mills theory has been proved to have zero radiative corrections and Ne'eman and Gell-Mann suggested in 1976 [47) that N = 8 Supergravity might produce a finite quantum gravity. Cremmer et al constructed the model in 1979-80 [48]. For a while there was a great interest in the theory (which was constructed as N = 1 in d = 11 dimensions and has to be compactified, but with the 1984 "rebirth" of String Theory, attention veered in that direction. However, in 1995 it was shown that this model can be obtained by truncation of a Membrane theory in 11 dimensions [49]. This "M-theory" has now replaced the String as the candidate "theory of everything" and its quantum field theory truncation is again at the center of the picture. Note that Yuval's interest in the quantum membrane preceded the present wave and his book (with E. Eizenberg) Membranes and Other Eztendons (p-branes) appeared in print just as the general interest was awakening [50].

xxv Back in 1977, Ne'eman and Tullio Regge [51] explored the basic geometric and algebraic structure of both gravity and supergravity and demonstrated that the local supersymmetric gauge transformations are in fact Lie derivatives, or anholonomized general coordinate transformations. A by-product of this study was the method of working on the group manifold, with spontaneous factorization (of the homogeneous Lorentz subgroup's manifold coordinates). Yuval further developed the technique with J. Thierry-Mieg. The latter work led to Thierry-Mieg's elegant geometrical interpretation of the BRS algebra [52]. Astronomy and Cosmology When the quasars were first identified as objects lying at cosmological distances, Ne'eman [53] and independently also Novikov, suggested that quasars are white holes, lagging cores from the cosmological expansion. With Gerald Tauber, he further developed the idea [54], assuming a deSitter mechanism (with quantum vacuum energy as a cosmological constant for the source-term). Quasars are now believed to represent large black holes instead, but the Ne'eman-Novikov-Tauber theory was a precise precursor of the presently favored eternal inflationarv cosmogony of Guth and Linde [55] except for the presently extremely larger scale. It is, in fact, used for the evaluation of the density fluctuations, often based on Harrison's analysis of the Ne'eman-Tauber models [56]. A prediction with astrophysical conclusions, which has yet to be realized, was made by Ne'eman (basing his analysis on S-matrix considerations) and by Bodmer (from nuclear theory considerations). It relates to the existence of hypercollapsed nuclear states [57] and may well soon be tested in heavy-ion collision experiments, intended to explore the "quark-gluon plasma." Note that Yuval's interest in Astronomy has found other realization channels. He was responsible for the creation of the Wise Observatory in the Negev in 1971 (with a 49 inch wide-angle telescope) and of the only department of Physics and Astronomy in Israel. Astronomers celebrated the Wise Observatory's 25th anniversary in 1996, relating the story of its birth [58]. Philosophy of Science Yuval has made several important contributions in the analysis of the sequence phenomenology, conservation features, classification, structure. Another, perhaps more important lesson from the practical aspect, has been his interpretation of the role of science in the evolution of human societies. Every evolutionary process requires (1) a randomized mutational mechanism and (2) a procedure selecting stable "good" mutations, making the series of evolutionary stages. In Biology, the randomized machinery corresponds to errors in the DNA copying mechanism and the levels are the sequence of species selected by environmental or (at the gene level) dynamical considerations. In nucleosynthesis, the randomized mutations occur in the high energy scattering of nuclei induced by gravitational pressures and the levels are the more stable nuclei. What are the corresponding factors in the evolution of human societies? The stable levels are characterized by technologies: the stone age, the bronze age, etc., and more recently, the industrial age, the nuclear age, the age of infor-

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mation technology. Where is then the randomized mutational procedure? Yuval's discovery is that it is the scientific research mechanism which does it. Any great discovery is by definition a surprise and cannot have been searched for. The day research will only go for "what we already know is there," that day will also be the day of stagnation for society. Any establishment which tries to overdirect its researchers is thereby stopping advance [59-60]. The examples abound and we shall soon have the possibility of reading the English translation of his 1996 Van Leer Lectures on this subject. This approach, generalizing evolution, also throws a new light on questions of morals and humanism. Yuval reanalyzed those aspects in Nietzsche's writings [61] which praise strength and despise pity and the weak, aspects which were used in Nazi ideology. They result from applying evolution at the wrong level: rather than survival of the fittest at the level of the individual this is survival of the fittest society-and a society is like a chain, its strength is that of the weakest link, which thus requires help and maintenance. That altruism has evolutionary advantages on the genetic level was already emphasized by W. Hamilton. Yuval extended the point at the societal level, with culture as the information encoding mechanism. Yuval has also investigated the evolutionary mechanism in science itself and has studied the role and characterization of serendipitous advances [62-63]. With Kantorowitz, he has developed the concept of evolutionary epistemology, proposed by Popper and Campbell [64]. Both of the above studies should serve as advised readings for whoever is charged with the responsibility for research programs. Yuval had the opportunity to apply his results-in his various capacities, whether as Scientific Director of the IAEC Soreq Establishment (1961-63), as IAEC Acting Chairman (1982-84,1990-92), as University President (1971-75), as Chief Scientist of the Defense Ministry (74-76), or as Minister of Science (1982-84, 1990-92), or Minister of Energy (1990-92) or also as Chairman of Israel's Space Agency (since 1983). It would be interesting to sort out the various relevant case histories ... In conclusion I would like to express our profound gratitude to Yuval Ne'eman for what he has done through the years, for our meetings, for Physics, for Science and Life not only in Israel but on the Planet Earth. Remo Ruffini References

[1]

[2] [3] [4] [5]

Y. N., "Derivation of Strong Interactions from a Gauge Invariance", Nucl. Phys. 26 (1961) p. 222-229. Reprinted in The Eightfold Way, p.58 (in colI. with M. Gell-Mann) "Frontiers in Physics" series, W. A. Benjamin Inc., New York, 1964; translated into Russian and reprinted in Elementary Particles and Gauge Fields, D. Ivanenko, ed., Mir. Pub., Moscow, p. 176 (1964). G. Goldhaber, in From SU(3) to Gravity, E. Gotsman and G. Tauber eds., Cambridge U. Press (1985), pp 103-106. V.E. Barnes et al., Phys. Rev. Lett. 12 (1964) 204. J. and M. Gribbin, Richard Feynman, a Life in Science, Dutton Books (Penguin) (1997), p. 192-194. El. J. Lipkin, Phys. Reports 8C (1973) 175, p. 180.

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[6] [7] [8] [9] [10] [11] [12]

[13]

[14]

[15] [16] [17] [18] [19] [20] [21]

[22] [23]

[24]

M. Gell-Mann, Phys. Rev. 125 (1962) 1067. M. Gell-Mann in Proc. Intern. Conf High En. Phys. (CERN 1962) p. 805. M. Gell-Mann and Y. N., The Eightfold Way, W.A Benjamin Pub., New York (1964), p. 317. Y. N., "The Fifth Interaction", Phys. Rev. B134 (1964) p. 1355-1357, also reprinted in The Eightfold Way, pp. 297-299 (see Books, No.1). Y. N., "Embedded Space-Time and Particle Symmetries", Rev. Mod. Phys. 37 (1965) p. 227-230. N. Cabibbo, L. Horwitz and Y. N., "The Algebra of Scalar and Vector Vertex Strengths in Regge Residues", Phys. Lett. 22 (1966) p. 386-340. Y. N., "Symmetries Related to the Quark Model", in Proc. of the Uppsala Fifth Intern Conf. on High Energy Physics and Nuclear Structure, G. Tibell, ed. (Alonquist and Wiksell, Stockholm, 1974), p. 10-2l. N. Cabibbo, J. J. J. Kokkedee, L. Horwitz and Y. N., "Possible Vanishing of Strong Interaction Cross-Section at Infinite Energies", Nuovo Cimento 45 (1966) p. 275-280. Y. Dothan and M. Gell-Mann and Y. N., "Series of Hadron Energy Levels as Representations of Non-Compact Groups", Phys. Lett. 17 (1965) p. 148-15l. Also reprinted in Symmetry Groups in Nuclear and Particle Physics, F. J. Dyson, ed., (W. A. Benjamin, New York, 1966), p. 283-286 (1966) and in Dynamical Groups and Spectrum Generating Algebras, A. Bohm and Y. N., eds., (World Scientific, Singapore, 1988), I p. 433-436; Y. Dothan and Y. N., "Band Spectra Generated by Non-Compact Algebra", in Resonant Particles (Proc. of the Second Athens Cont on Resonant Particles), B. A. Munir, ed., (Ohio University, Athens, Ohio, 1965), p. 17-32, also reprinted in Symmetry Groups in Nuclear and Particle Physics, F. S. Dyson, ed., (W. A. Benjamin, New York, 1966), p. 287-310. G. Veneziano, Phys. Reports C9 (1974) 199. A. Barut, A. Bohm and Y. N., eds., Dynamical Groups and Spectrum Generating Algebras, World Scientific Pub., Singapore (1988), 2 vols., p. 1138. D. W. Joseph, Y. N. (1969) unpub. See also D. W. Joseph, University of Nebraska preprint (1970), unpub. Dj. Sijacki, J. Math. Phys. 16 (1975) 298. L. C. Biedenbarn et al., Phys. Lett. B42 (1972) 257. Y. N. and Dj. Sijacki, "QCD as an Effective Strong Gravity", Phys. Lett. B247 (1990) p. 571-575. Y. N. and Dj. Sijacki, "Proof of Pseudo-Gravity as QCD Approximation for Hadron IR Region and J '" M2 Regge Trajectories" , Phys. Lett. B276 (1992) p.173-178. Y. N. and Dj. Sijacki, "Chromogravity: I QCD-Induced Diffeomorphisms", Inter. Jour. Mod. Phys. AI0, No. 30 (1995) p. 4399-4412. Y. N. and Dj. Sijacki, "Inter-hadron QCD-induced Diffeomorphisms, from a Radial Expansion of the Gauge Field", in Mod. Phys. Lett. All (1996) p.217-225. See examples cited in ref. 7 of Y. N. and Dj. Sijacki, "GL(4,R) GroupTopology, Covariance and Curved-Space Spinors", Int. J. of Mod. Phys. A2

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(1987) p. 1655-1669. [25) F.W. Hehl, O.D. Kerlick and P. v.d. Heyde, Phys. Lett. B63 (1976) 446. [26) Y.N., p. W. Hehl and E. A. Lord, "Hypermomentum in Hadron Dynamics and in Gravitation", Phys. Rev. DI7 (1978) p. 428-433. [27) Y. N., "Gravitational Interaction of Hadrons: Band-Spinor Representations of GL(n,R)", Proc. Nat. Acad. Sci. (US) 74 (1977) p. 4157-4159. [28) Y. N., "Spinor-Type Fields with Linear, Affine and General Coordinate Transformations", Annales de l'Institut Henri Poincare 28 (1978) p. 369-378; Dynamical Groups and Spectrum Generating Algebras, A. Bolim and Y. N., eds., World Scientific, Singapore (1988), II, p. 846-855. [29) F. Budinich and A. Trautman, The Spino rial Chessboard, Springer Verlag (1988) p. 3. [30) D. Finkelstein, The Structure of Spacetime, Springer Verlag (1997) p. 356. [31) A. Cant and Y. N., "Spinorial Infinite Equation Fitting Metric Affine Gravity", J. Math. Phys. 26 (1985) p. 3180-3189. [32) Dj. Sijacki and Y. N., "Algebra and Physics of the Unitary Multiplicity Free Representations of SL(4,R)", J. Math. Phys. 26 (1985) p. 2457-2464, and in Dynamical Groups and Spectrum Generating Algebras, A. Bolim and Y. N., eds., World Scientific, Singapore (1988), II, p. 808-815. [33) Y. N. and Dj. Sijacki, "SL(4,R) World Spinors and Gravity", Phys. Lett. 157B (1985) 275-279, and in Dynamical Groups and Spectrum Generating Algebras, A. Bohm and Y. N., eds., World Scientific Singapore, 1988), II, p.873-877. [34) F. W. Hehl, J. D. McCrea, E. Mielke, and Y. N., "Progress in Metric-Affine Gauge Theories of Gravity with Local Scale Invariance", Found. Phys. 19 (1989) p. 1075-1100. [35) Y. N., F. W. Hehl and B. A. Lord, "Hadron Dilation, Shear and Spin as Components of the Intrinsic Hypermomentum Current and Metric-Affine Theory of Gravitation" , Phys. Lett. 71B (1977) p. 432-434. [36) F. W. Hehl, J. Dermott McCrea, E. W. Mielke and Y. N., "Metric-Affine Gange Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance", Phys. Rep. 258 (1995) p. 1-171. [37) Y. N. and Dj. Sijacki, "Gravity from Symmetry Breakdown of a Gauge Affine Theory," Phys. Lett. B200 (1988) p. 489-494. [38) C. Y. Lee and Y. N., "Renormalization of Gauge Affine Gravity", Phys. Lett. B242 (1990) p. 59-63. [39) Y. N., "Irreducible Gauge Theory of a Consolidated Salam- Weinberg Model", Phys. Lett. B8I (1979) p. 190-194; D. B. Fairlie, Phys. Lett. B82 (1979) 97. [40) V. N. and J. Thierry-Mieg, "Geometrical Gauge Theory of Ghost and Goldstone Fields and of Ghost Symmetries", Proc. Nat. Acad. Sci. (USA), 77 (1980) p. 720-723. [41) D. Quillen, Topology 24 (1985) 89; Y. N. N212. [42) R. Coquereaux et al., Int. J. Mod. Phys. A7 (1992) 2809,6555. [43) A. Connes and J. Lott, Nucl. Phys. (Proc. Suppl.) BI8 (1990) 29. [44) L. Corwin, S. Sternberg, and Y. N., "Graded Lie Algebras in Mathematics and Physics, (Bose-Fermi Symmetry)", Rev. Mod. Phys. 47 (1975) p. 573-604.

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[45] V. G. Kac, Func. Analys. Appl. 9 (1975) 9l. [46] D.Z. Freedman et aI, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. B62 (1976) 335. [47] Y. N., June 76 lecture at Aspen Physics Institute (unpublished). [48] E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 14l. [49] E. Bergshoeff, E. Sezgin and P.K Townsend, Phys. Lett. 189 (1987) 75 and B209 (1988) 45l. [50] Y. N. and E. Eizenberg, Membranes and Other Extendons (p-branes), World Scientific, Singapore (1995). [51] Y. N. and T Regge, "Gauge Theory of Gravity and Supergravity on a Group Manifold", Riv. Nuovo Cim., 1 #5 (Series 3) (1978), p. 1-42 (First issued as lAS Princeton and U Texas ORO 3992 328 Preprints). [52] Y. N., T. Regge and J. Thierry-Mieg, "Models of Extended Supergravity as Gauge Theories on Group Manifolds" , Abstract for XIX Intern Conf. on High Energy Physics (Tokyo, 1978). [53] Y. N., "Expansion as an Energy Source in Quasi-Stellar Radio Sources", Astrophys. J. 141 (1965) p. 1303-1305. [54] Y. N. and G. Tauber, "The Lagging-Core Model for Quasi-Stellar Sources", Astrophys. J. 150 (1967) p. 755-766. [55] A. Guth, Phys. Rev. D23 (1981) 387; A. Starobinsky, Phys. Lett. B91 (1980) 99; A.D. Linde, Inflation and Quantum Cosmology, Academic Press, Boston (1990). [56] E. Harrison, Phys. Rev. Dl (1970) 2726; see ref. 26. [57] Y. N., "Nuclear Physics Implication of the Spin 2 Multiplet", Symmetry Principles at High Energy, (Proc. of the Fifth Coral Gables Conf. (1968)), W. A. Benjamin, New York (1968), p. 149-15l. [58] Y. N., "Renewal of the Astronomical Research in Eretz- Israel (25 Years of the Florence and George Wise Observatory) (Hebrew) Bull. Israel Acad. Sci. of Humanities 2 (Feb. 1997) p. 8-12. [59] Y. N., "Science as Evolution and Transcendance", Proc. Fairchild Symp. on the Relevance of Science, Pasadena (1977), Acta Scientijica Venezolana 31 #1-3 (1980); French translation in Concordances #18 (1978) p. 14-16. Modified Spanish version in Naturaleze Mezico 11 (1980) p. 16-20. [60] Y. N., The Evolutionary Role of Research", A. Romano, ed., in Metabolic, Pediatric and Systemic Opthalmology, Pergamon Press Limited, USA/UK (1988) 11, p. 12-13. [61] Y. N., J. Social and Evolutionary Systems 15 (1992) 347. [62] A. Kantorovich and Y. N" "Serendipity as a Source of Evolutionary Progress in Science", Studies in History and Philosophy of Science 20 (1989) p. 505529. [63] Y. N., "Serenipity, Science and Society-An Evolutionary View", Proc. Kon. Ned. Akad. v. Wetensch. 96 #4 (1993) p. 433-448. (Emil Starken-stein Symposium Invited Lecture, Rotterdam 1992). [64] D. T. Campbell, in Studies in the Philosophy of Biology, F.J. Ayala and T. Dobzhansky, eds., Macmillan, London (1974) p.139-161; also in The Philosophy of Karl Popper, P.A. Scilpp ed., La Salle (1974) p. 413-463.

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CONTENTS . . . . .

v

Organizing Committees and Sponsors

vii

Marcel Grossmann Awards

ix

Publications in this Series

xiii

Preface Dedication in Honour of Yuval Ne'eman's 70th Birthday

xv

PART A

PLENARY SESSIONS Discrete Gravity TULLIO REGGE .

2

The Quantum and Gravity: The Wheeler-DeWitt Equation BRYCE DEWITT . . . . . . . . . . . . . . . . .

6

Perturbative Dynamics of Quantum General Relativity JOHN F. DONOGHUE . . . . . . . . . . . . . . .

26

Broken Symmetry: Applying the Method of the Superconnection for Riemannian Gravity YUVAL NE'EMAN . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Recent Developments in String Theory ASHOKE SEN . . . . . . . . . . . .

47

Black Holes and D-Branes JUAN M. MALDACENA . .

55

The Notions of Time and Evolution in Quantum Cosmology R. PARENTANI . . . . . . . . . . . . . . . . . . . .

76

Quantum Black Holes as Atoms JACOB D. BEKENSTEIN

92

Curvature-Based Hyperbolic Systems for General Relativity YVONNE CHOQUET-BRUHAT, JAMES W. YORK, JR., ARLEN ANDERSON

112

Inner Structure of Spinning Black Holes AMOS ORI . . . . . . . . . . . . .

122

Has the Black Hole Equilibrium Problem Been Solved? B. CARTER . . . . . . . . . . . . . . . . . . . .

136

Progress in the Development of Resonant Mass Gravitational Wave Detectors D.G. BLAIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

156

xxxii Laser Interferometric Gravitational Wave Detectors Spanning 7 Decades of Frequency K.A. STRAIN et al. . . . . . . . . . . . . . . . . . . . . . . . . .

176

Virgo Status Report - July 1998 PIERO RAPAGNANI et al. . . . .

194

Water Masers in Active Galactic Nuclei M. INOUE . . . . . . . . . . . . .

197

Astrophysical Evidence for Black Hole Event Horizons KRISTEN MENOU, ELIOT QUATAERT, RAMESH NARAYAN.

204

Large Scale Structure Lurz NICOLACI DA COSTA . . . . . . . . . . . . . .

225

Variations of the Cosmic Expansion Field and the Value of the Hubble Constant G.A. TAMMANN . . . . . . . . . . . . . . . . . . . . . . . . . . .

243

Data Reduction, Error Analysis and Identification of Systematic Errors in the Gravity Probe B Experiment M.1. HEIFETZ, C.W.F. EVERITT, G.M. KEISER, A.S. SILBERGLEIT . . . . .

259

PARALLEL SESSIONS Conservation Laws Chairperson: J. Katz Brief Survey on Conservation Laws J. KATZ . . . . . . . . . . . .

271

Conservation Laws for Cosmological Perturbations and Mach's Principle J. BICAK, J. KATZ, D. LYNDEN-BELL . . . . . . . . . . . . .

277

A Direct Method for Obtaining the Differential Conservation Laws A.A. CHERNITSKII . . . . . . . . . . . . . . . . . . . . .

280

From Taub Numbers to the Bondi Mass E.N. GLASS . . . . . . . . . . .

283

Machian Thought Experiments D. LYNDEN-BELL, J. KATZ, J. BICAK

286

The ADM Mass of Quasi-Asymptotically Flat Spacetimes ULISES NUCAMENDI, DANIEL SUDARSKY . . . . . . . .

288

On Superpotentials in General Relativity A.N. PETROV . . . . . . . . . . .

291

Conservation Laws in General Relativity ROSTISLAV F. POLISHCHUK . . . . .

294

xxxiii

The First Satisfactory Definition of Angular Momentum at Null Infinity ANTHONY RIZZI

....................... .

297

Conservation Laws in Cosmology JEAN-PHILIPPE UZAN

300

.....

Nonlocal Conservation Laws Derived from an Explicit Equivalence Principle R. A. VERA

......................... .

303

Exact Solutions Chairperson: I. Robinson The Lanczos Potential for the Weyl Tensor

307

F. ANDERSSON, S.B. EDGAR, A. HOGLUND, G. BERGQVIST .

Robinson-Trautman Radiative Space-Times with Cosmological Constant J. BICAK, J. PODOLSKY

. . . . . . .

. . . .

. . . . . .

310

. .

Abrupt Changes in the Multipole Moments of a Gravitating Body G.F. BRESSANGE, C. BARRABES, P.A. HOGAN

313

. . . . . . . .

Fluid Spheres of Uniform Density with Variable Space Curvature V.V. BURLIKOV, M.P. KORKINA

.

. .

316

. . . . . . . . . . .

Physical Interpretation of Cylindrically Symmetric Gravitational Fields Y. VERBIN ........... .

J. COLDING, N.K. NIELSEN,

319

Exact Wave Solution of Einstein's Equations V.1. DENISOV, V.V. GERSHOV

. . . .

322

. .

New Conformally Flat Radiation Metrics by GHP Integration S.B. EDGAR, G. LUDWIG.

. . . . .

. .

324

. . . . . . . . .

Toroidal Solutions to the Problem of Inhomogeneous Rotating, Gravitating Systems S. FILIPPI, A. SEPULVEDA

. . .

. . .

. . . . . . . . . . . .

. . . .

327

On Some Group Properties of Structure Equations of Newtonian Static Stars in Radiative and Convective Equilibrium ZDZISLAW A. GOLDA, MAREK SZYDLOWSKI

............. .

330

Modeling Dilaton Stars with Arbitrary Electromagnetic Field FRANCISCO S. GUZMAN M., TONATIUH MATOS C.

.....

Stationary Axisymmetric Dust Disks and Hyperelliptic Riemann Surfaces C. KLEIN, O. RICHTER ..................... .

333 336

Relativistic Disks as Sources of Kerr-Newman Fields

339

TOMAS LEDVINKA, MARTIN ZOFKA, JlRi BICAK

Nonholonomic Approach to Rotating Matter in General Relativity MATTIAS MARKLUND, GYULA FODOR, ZOLTAN PERJES

.....

342

xxxiv

Thermodynamical Stability of Relativistic Stellar Systems M. MERAFINA

. . . . . . . . . . . . . . . . . . .

345

How Much Quasinormal Ringing in a Gravitational Wave? H.-P. NOLLERT . . . . . . . . . . . . . . . . . . .

348

A New Approach to Schwarzschild Perturbations MARIO NOVELLO, JOSE MARTINS SALIM, MARTHA CHRISTINA MOTTA DA SILVA, RENATO KLIPPERT

. . . . . . . . . . . . . . . . . . . . . . .

351

Similarity Reduction for a Class of Algebraically Special Perfect Fluids A. RAINER, H. STEPHANI

.. . . . . . . . . . . . . . . . . . .

Two-Lambda Spherically Symmetric Solution B. SOLTYSEK, I. DYMNIKOVA . . . . . . . .

354 357

Test Particles and Photons in the Kerr-Newman-De Sitter Dyon Spacetimes Z. STUCHLlK . . . . . . . . . . . . . . . . . . . . . . . . . . . .

360

Ideal Gas Sources for the Tolman-Lemaitre-Bondi Metrics ROBERTO A. SUSSMAN

.............. .

363

Point Symmetries of the Field Equations for Type III Spaces with at Least One Killing Vector EDWARD P. WILSON. . . . . . . . . . . . . . . . . . . . . . . . .

366

Classical and Quantum Aspects of Radiation Fluid Shells: Black Holes and Wormholes, Mass Amplification, Light Dark Matter K.G. ZLOSHCHASTIEV . . . . . . . . . . . . . . . . . . . . . . . .

369

Monopole and Electrically Charged Dust Thin Shells in General Relativity: Classical and Quantum Comparison of Hollow and Atom-like Configurations K.G. ZLOSHCHASTIEV . . . . . . . . . . . . . . . . . . . . . . • .

372

Inertial Forces in General Relativity Chairperson: Robert T. Jantzen The Inertial Forces/Test Particle Motion Game DONATO BINI, PAOLO CARINI, ROBERT T. JANTZEN

376

Geometric Approach to the Split and Propagation of Light Rays STANISLAW L. BAZAN-SKI.

. . . . . . . . . . . . . . . . .

Interplay between Forces in Kerr-Newman Field J. BleAK, O. SEMERAK . . . . . . . . . . .

398 401

Gravitoelectromagnetism and Motion of Spinning Test Particles in General Relativity D. BINI, G. GEMELLI, R. RUFFINI.

. . . . . . . . . . . . . . . . . .

Circular Orbits in the Kerr Spacetime: Centrifugal Forces and Anti-Newtonian Behavior DONATO BINI, ROBERT T. JANTZEN, ANDREA MERLONI . . . . . . . . . .

404

407

xxxv Dirac Fermions in Almost Uniformly Accelerated Frames C. DARIESCU, M.A. DARIESCU

. .

410

. . . . . . . . . .

Equations of Motion of Spinning Relativistic Particle in External Fields I.B. KHRIPLOVICH, A.A. POMERANSKY

. . . . . . . . . . .

. . .

412

A Gravitational Kaleidoscope

413

ROBERTO SORIA, FERNANDO DE FELICE

Penrose Processes and the Gravitomagnetic Field REVA KAY WILLIAMS

416

........... .

Principle of Restricted Covariance in General Relativity

419

ROUSTAM ZALALETDINOV, REZA TAVAKOL, GEORGE ELLIS

Alternative Theories in Four Dimensions Chairperson: Friedrich W. Hehl Alternative Gravitational Theories in Four Dimensions FRIEDRICH W. HEHL

423

.............. .

Spinning Particle as Superblackhole

433

A.YA. BURINSKII . . . . . . . .

On the Non-Riemannian Manifolds as Framework for Geometric Unification Theories: Affine Connection Geometry with Asymmetric Metric SABRINA CASANOVA, GIOVANNI MONTANI, REMO RUFFINI, ROUSTAM ZALALETDINOV . . . . . . . . . . . . . . . .

436

.

Tensor Expressions in Rosen's Theory of Gravitation I.P. DENIS OVA

. . . . . . . .

439

. . . . . . . . .

The Geometry of Deformation Quantization and Self-Dual Gravity HUGO GRAciA-COMPEAN, JERZY F. PLEBANSKI, MACIEJ PRZANOWSKI

442

Scalar Field from Dirac Coupled Torsion RICHARD T. HAMMOND, CHARRO GRUVER, P.F. KELLY

445

Is a 'Hadronic' Shear Current One of the Source in Metric-Affine Gravity? FRIEDRICH W. HEHL, YURI N. OBUKHOV . . . . . . . . . . .

448

Bianchi Cosmology with Spinning Matter and Magnetic Field in the Einstein-Cartan Theory K.H. KONG, K.S. CHENG, P.C.W. FUNG, H.Q. Lu. . . . . . . .

451

New Constraints on Space-Time Torsion from Hughes-Drever Experiments C. LAMMERZAHL

....................... .

454

Can the Electromagnetic Field Couple to Post-Riemannian Structures? C. LAMMERZAHL, R.A. PUNTIGAM, F.W. HEHL . . . . . .

. . .

. .

457

Space-time Averages in Macroscopic Gravity MARC MARS, ROUSTAM M. ZALALETDINOV

460

xxxvi General Relativity as Continuum with Microstructure. A Framework Based on Formal Theory of Lie Pseudogroups A.Yu. NERONOV

. . . . . . . . . . . . . . . . . . . . . . . . . . "

463

On a Theory for Nonminimal Gravitational-Electromagnetic Coupling Consistent with Observational Data R. OPHER, U.F. WICHOSKI. . . . . . . . . . . . . . . . . . . . .

466

Torsion, Thermodynamical, Quantum and Hydrodynamical Fluctuations D.L. RAPOPORT

....................... .

469

General Relativity in Terms of Dirac Eigenvalues C. ROVELLI

............... .

472

Unification of Gravitation and Electroweak Interactions RUGGERO MARIA SANTILLI

........... .

473

An Alternative Perspective in Quantum Mechanics and General Relativity B.G. SIDHARTH . . . . . . . . . . . . . . . . . . . . . . . . . .

476

Fluctuational Cosmology B.G. SIDHARTH . . .

479

Towards Hypergravity DJORDJE SIJACKI . .

482

Newton's Precession Theorem, Eccentric Orbits and Mercury's Orbit S.R. V ALLURI, W.L. HARPER, R. BIGGS . . . . . . . . . . . . .

485

Gravitation: Field and Curvature L.V. VEROZUB

. . . . . . . .

489

Gravitation Spin Effect and Magnetic Inclination of Pulsars C.M. ZHANG, K.S. CHENG,

X.J. Wu

....... .

492

Alternative Theories (Kaluza-Klein) Chairperson: V.N. Melnikov Geometric Extended Gravity Theory of Yang-Mills Type M.F. BORGEs, S.R.M. MASALSKIENE

. . . . . . . . .

496

Gravitating P-Brane Systems, Black Holes and Wormholes K.A. BRONNIKOV . . . . . . . . . . . . . . .

499

Reasons for the Space-Time to be Four-Dimensional F. BURGBACHER, C. LAMMERzAHL, A. MACIAS . .

502

Mathematical Problems in Higher Order Gravity and Cosmology S. COTSAKIS . . . . . . . . . . . . . . . . . . . . . . .

505

Multidimensional Geometrical Model of the Electrical and SU(2) Colour Charge with Splitting Off the Extra Coordinates V.D. DZHUNUSHALIEV . . . . . . . . . . . . . . . . . . . . . . . . . .

508

xxxvii

Multidimensional SU(2) Wormhole between Two Null Surfaces V.D. DZHUNUSHALIEV . . . . . . . . . . . . . . . . . .

511

Gravitational Theory without the Cosmological Constant Problem E.!. GUENDELMAN, A.B. KAGANOVICH ........ .

514

Gravitational Excitons from Extra Dimensions U. GUNTHER, A. ZHUK . . . . . . . . . .

517

An Alternative KK Theory and Black Hole Formation LEOPOLD HALPERN . . . . . . . . . . . . . . .

520

Multidimensional Gravity with P-Branes V.D. IVASHCHUK, V.N. MELNIKOV

523

Complete Classification of 1+1 Gravity Solutions T. KLOSCH, T. STROBL . . . . . . . . . . .

526

Hairy Black Holes Chairperson: D. Maison Summary Report on the Session on Hairy Black Holes D. MAISON . . . . . . . . . . . . . . . . . . .

530

Extreme Black Holes Won't Wear Long Hair A. CHAMBLIN, J.M.A. ASHBOURN-CHAMBLIN, R. EMPARAN, A. SORNBORGER . . . . . . . . . . . . . . . . . . .

536

Singularities Inside Hairy Black Holes D.V. GAL'TSOV, E.E. DONETS, M.Yu. ZOTOV

539

Mass Formulas for a Class of Stationary Black Holes M. HEUSLER . . . . . . . . . . . . . . . . . .

542

Static Regular and Black Hole Solutions with Axial Symmetry in EYM and EYMD Theory B. KLEIHAUS, J. KUNZ. . . . . . . . . . . . . . . . . . . . . . . .

545

The Non-Linear Sigma Model on De Sitter Space CH. LECHNER, P.C. AICHELBURG . . . . . . .

548

Limiting Solutions of Sequences of Globally Regular and Black Hole Solutions in SU(N)-EYMD Theories A. SOOD, B. KLEIHAUS, J. KUNZ . . . . . . . . . . . . . . . . . . . .

551

Spherical Black Holes Cannot Support Scalar Hair DANIEL SUDARSKY, THOMAS ZANNIAS. . . . .

554

Non-Abelian Black Holes in Brans-Dicke Theory T. TORII, T. TAMAKI, K. MAEDA . . . . . . .

557

On Stationary Black Holes of the Einstein-Conformally Scalar Field System T. ZANNIAS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

560

xxxviii Time Machines Chairperson: A. Carlini Time Machines

564

A. CARLINI

The Principle of Self-Consistency and the Cauchy Problem for a Self-Interacting Relativistic Particle in the Presence of a Time Machine A. CARLINI, I.D. NOVIKOV . . . . . . . . . . . . . .

. . . . . . • . ..

584

Time Machine and Foliations A.K. GUTS . . . . . . . .

587

Causality in Topologically Nontrivial Space-Times Yu. KONSTANTINOV . . . . . . . . . . . .

590

M.

Time Machines with Non-Compactly Generated Cauchy Horizons and "Handy Singularities" S.V. KRASNIKOV

. . . • . .

. . . . . . . . . . .

593

On Causality Violation and Singularities KENGO MAEDA, AKIHIRO ISHIBASHI, MAKOTO NARITA

596

Lorentzian Wormholes in a De Sitter Cosmos

F.

SCHEIN

...............•....

599

On the Causal Structure of Spinning Einstein-Yang-Mills Strings R.J. SLAGTER

. . • . . . • .

. . . . . . . . . . . . • .

602

Gravity and Time Machines C.S. UNNIKRISHNAN.

605

.

The Reliability Horizon

608

MATT VISSER

Chaos in General Relativity Chairperson: V. Gurzadyan On the Thrbulence near Cosmological Singularity V. BELINSKI, A. KIRILLOV, G. MONTANI.

. ..

612

Fractals and Symbolic Dynamics as Invariant Descriptors of Chaos in General Relativity N.J. CORNISH

............................

613

The Mixmaster Universe is Unambiguously Chaotic N.J. CORNISH, J.J. LEVIN

. . .

. . . . . . . .

Chaotic Exit to Inflation: The Dynamics of Pre-Inflationary Universes H.P. DE OLIVEIRA, I. DAM lAO SOARES, T.J. STU CHI . . . . . . .

616

619

On the Stochasticity in the Mixmaster Model A.A. KIRILLOV, G. MONTANI . . . . . . . . . . .

622

xxxix Chaotic Exit to Inflation: Pre-Inflationary Friedmann-Robertson-Walker Universes G.A. MONERAT, H.P. DE OLIVEIRA, I. DAML4.0 SOARES . . . . . . .

625

Perturbation Theory in Macroscopic Gravity: On the Definition of Background GIOVANNI MONTANI, ROUSTAM ZALALETDINOV. . . . . . . . . . . . . .

628

Chaos and Quantumlike Mechanics in Atmospheric Flows: A Superstring Theory for Supergravity A. MARY SELVAM . . . . . . . . . . . . . . . . . . . . . . . . .

631

Chaos in Schwarzschild and Kerr Space time - The Motion of a Spinning Particle SHINGO SUZUKI, KEI-ICHI MAEDA. . . . . . . . . . . . . . . . . .

634

Relaxations and Long Time Evolutions of One-Dimensional N-Body Systems T. TSUCHIYA, N. GOUDA, T. KONISHI . . . . . . . . . . . . . . . . .

637

Einstein Maxwell Systems Chairperson: Chul H. Lee New Two-Component Spinor Formulae for Classical General Relativity J.G. CARDOSO . . . . . . . . . . . . . . . . . . . . . . . . .

641

Cauchy Problem for a Relativistic Charged Continuum CARLO CATTANI, ETTORE LASERRA . . . . . . . .

644

Electrostatic Field Generation in the Pulsar Magnetosphere Relativistic Plasma O.V. CHEDIA, T.A. KAHNIASHVILI, G.Z. MACHABELI, I.S. NANOBASHVILI

647

Split Structures in General Relativity V.D. GLADUSH, R.A. KONOPLYA . .

650

Torsional Weyl-Dirac Electrodynamics M. ISRAEL IT . . . . . . . . . . .

653

Can Non-Gravitational Black Holes Exists? MARIO NOVELLO, VITORIO A. DE LORENCI, EDGAR ELBAZ

656

On Scattering Off the Extreme Reissner-Nordstrom Black Hole in N = 2 Supergravity T AKASHI OKAMURA. . . . . . . . . . . . . . . . . . . . . . .

659

Knots in Simulations of Magnetized Relativistic Jets MAURICE H.P.M. VAN PUTTEN . . . . . . . . . .

662

Mass Inflation Chairperson: Eric Poisson Parallel Session on Mass Inflation ERIC POISSON . . . . . . . .

666

xl

Mass Inflation Inside Non-Abelian Black Holes P. BREITENLOHNER, G. LAVRELASHVILI, D. MAISON

670

Structure of the Cauchy Horizon Singularity LIOR M. BUR1W . . . . . . . . .

673

Some Cosmological Tails of Collapse C.M. CHAMBERS, P.R. BRADY, W. KRIVAN, P. LAGUNA

676

Oscillatory and Power-Law Mass Inflation in Non-Abelian Black Holes D.V. GAL'TSOV, E.E. DONETS, M.Yu. ZOTOV . . . . . . . . . . .

679

Mass-Inflation in Dynamical Gravitational Collapse of a Charge Scalar-Field S. HOD, T. PIRAN. . . . . . . . . . . . . . . . . . . . . . . . . .

682

Stability of Plane-Wave Cauchy Horizons D.A. KONKOWSKI, T.M. HELLIWELL

685

Critical Phenomena Chairperson: Patrick R. Brady Critical Phenomena in Gravitational Collapse PATRICK R. BRADY, MIKE J. CAl . . . . . .

689

Spherical Collapse of a Mass-Less Scalar Field With Semi-Classical Corrections S. AYAL, T. PIRAN . . . . . . . . . . . . . . .

705

A Critical Look at Massive Scalar Field Collapse C.M. CHAMBERS, P.R. BRADY, S.M.C.V. GONQALVEZ

708

Dynamical Instability and Critical Behavior of van Putten's "Approximate Black Hole" MATTHEW W. CHOPTUIK, ERIC W. HIRSCHMANN, STEVEN L. LIEBLING.

711

Critical Behaviour and Universality in Gravitational Collapse of a Charged Scalar Field S. HOD, T. PIRAN. . . . . . . . . . . . . . .

714

Fine-Structure of Choptuik's Mass-Scaling Relation S. HOD, T. PIRAN. . . . . . . . . . . . . . .

717

The Structure of Singularity in Gravitational Collapse S. JHINGAN . . . . . . . . . . . . . . . . . .

720

Quantum Mass Gap at the Threshold of Black Hole Formation I LEONARD PARKER, YOAV PELEG . . . . . . . . . . . . . .

723

Quantum Mass Gap at the Threshold of Black Hole Formation II YOAV PELEG, LEONARD PARKER . . . . . . . . . . . . . .

725

xli

Numerical Relativity Chairperson: J.-A. Marek Gauge Schocks in Hyperbolic Relativity MIGUEL ALCUBIERRE, JOAN MASSO . .

729

Evolutions of Stellar Oscillations G.D. ALLEN, N. ANDERSSON, K.D. KOKKOTAS, B.F. SCHUTZ.

732

Lattice Models of 2D Quantum Gravity E. BITTNER, A. HAUKE, H. MARKUM, J. RIEDLER, C. HOLM, W. JANKE.

735

BH Punctures as Initial Data for General Relativity S.R. BRANDT, B. BRUGMANN . . . . . . . . .

738

Three Dimensional Distorted Black Holes: Initial Data and Evolution S.R. BRANDT, K. CAMARDA, E. SEIDEL . . . . . . . . . . . . .

741

The Matter Plus Black Hole Problem in Axisymmetry S.R. BRANDT, J .A. FONT . . . . . . . . . . . .

744

New Coordinate Systems for Axisymmetric Black Hole Collisions S.R. BRANDT, P. WALKER, P. ANNINOS . . . . . . . . . . .

747

Free Evolution of Nonlinear Scalar Field Collapse in Double-Null Coordinates LIOR M. BURKO . . . . . . . . . . . . . . . . . . . . . . . . . . .

750

Variational Method and Regge Equations CARLO CATTANI . . . . . . . . . . .

753

A 1D Solver for the Einstein Field Equations J.A. FONT, J. MASSO . . . . . . . . . .

756

Black Holes Illuminated by Infalling Material A.GOMBOC,C.FANTON,L.CARLOTO,A.CADEZ

759

Graph Theory in Mathematical Cosmology R. MONDAINI, F. MONTENEGRO

762

Gravitational Radiation from Colliding Black Holes: The Close Limit Approximation H.-P. NOLLERT . . . . . . . . . . -. . . . . . . . . . . . . .

765

Nonlinear Wave Equations for Numerical Relativity: Towards the Computation of Gravitational Wave Forms of Black Hole Binaries Maurice H.P.M. VAN PUTTEN

768

Newtonian and Post-Newtonian Binary Neutron Star Mergers HISA-AKI SHINKAI, WAI-Mo SUEN, F. DOUGLAS SWESTY, MALCOLM TOBIAS, EDWARD Y.M. WANG, CLIFFORD M. WILL. . . . . . . . . . . . . . . .

771

Lorentzian Dynamics in the Ashtekar Gravity HISA-AKI SHINKAI, GEN YONEDA . . . . . .

774

xlii

Algebraic Computations Chairperson: Seryei A. Klioner EinS: A Mathematica Package for Computations with Indexed Objects SERGEI A. KLIONER . . . . . . . . . . . . . . . . . . . . . . .

778

Tensign, the Philosophy Behind a New Userinterface A. HOGLUND . . . . . . . . . . . . . .

781

Quantum Fields in Curved Space Time Chairperson: V.I. Belinskii Quantum Fields in Anti-De Sitter Wormholes C. BARCELO, L.J. GARAY . . . . . . . . .

785

On the Theory of the Unruh Effect V.A. BELINSKII, B.M. KARNAKOV, V.D. MUR, N.B. NAROZHNYI

788

Somewhat Managable Propagators in Curved Space KARTEN BORMANN, FRANK ANTONSEN . . . . . .

791

Semiclassical Quantization on Black Hole Spacetimes R. CASADIO, B. HARMS, Y. LEBLANC . . . . . . .

794

Instructive Properties of Quantized Gravitating Dust Shell A.D. DOLGOV, I.B. KHRIPLOVICH . . . . . . . . . . .

797

Gauge Dependence of the Effective Average Action in Einstein Gravity S. FALKENBERG, S.D. ODINTSOV . . . . . . . . . . . . . . . . .

800

Transition Amplitudes of Interacting Charged Particles in an Electric Field and the Unruh Effect CL. GABRIEL, PH. SPINDEL, S. MASSAR, R. P ARENTANI . .

803

Acceleration-Induced Carrier of the Imprints of Gravitation ULRICH H. GERLACH . . . . . . . . . . . . . . . . .

806

Gravitational Conical Bremsstrahlung and Differential Structures J. GRUSZCZAK . . . . . . . . . . . . . . . . . . . . .

809

Radiation by Static Sources Outside Static Black Hole? ATsusHI HIGUCHI, GEORGE E.A. MATSAS, DANIEL SUDARSKY

812

Superluminal Velocities of Photons in Gravitational Background LB. KHRIPLOVICH . . . . . . . . . . . . . . . . . . . .

815

Kinematics and Uncertainty Relations of a Quantum Test Particle in a Curved Space-Time P. KUUSK, J. ORD

818

Particle Creation and Vacuum Polarization of Nonconformal Scalar Field Near the Isotropic Cosmological Singularity J. LINDIG, V.M. MOSTEPANENKO . . . . . . . . . . . . . . . . . . . .

821

xliii

Metric Fluctuations in Semiclassical Gravity R. MARTIN, E. VERDAGUER . . . . . . .

824

Energy-Momentum Tensor and Particle Creation in the De Sitter Universe CARMEN MOLINA-P ARfs . . • . . . . . . . . . . . • . . . . . . .

827

Scalar Perturbations in Open Semiclassical FLRW Universe G. SIEMIENIEC-OZII~BLO, A. WOSCZYNA . . . . . . . . .

830

Classical Backgrounds with Vanishing Effective Lagrangians L. SRIRAMKUMAR, T. PADMANABHAN, R. MUKUND . . . .

833

An Approach to Quantum Statistical Mechanics on Robertson-Walker Space-Times

836

M. TRUCKS

Spinning Particles on Curved Spaces and Constants of Motion M. VISINESCU

•.

. .

. . . . . . . • . . . . • . . . .

839

Gravitational Vacuum Polarization MATT VISSER

842

........ .

PARTB Time in Quantum Gravity Chairperson: G. Horwitz Thermodynamics for Gravitating Systems is Different

G.

HORWITZ

................. .

The Evolution Operator for Quantum Gravity: Signature Change in Minisuperspace A. CARLINI, H. ISHIHARA, K. NAKAMURA, T. OKAMURA. . . . . .

846

862

Kolmogorov's Algorithmic Complexity and its Probability Interpretation in Quantum Gravity V.D. DZHUNUSHALIEV . . . . . . . . . . . . . . . . . . . . . . . .

865

Covariant Introduction of the Planck Length in the Path Integral Formulation of Field Theory ALESSANDRO MASSAROTTI, ALAK CHAKRAVORTY

868

Quantum Reference Frames and Quantum Space-Time S.N. MAYBUROV

. . . . .

. .

. . . . . .

871

Time in Quantum Gravity C. ROVELLI

..... .

874

Time, Gauge, and the Superposition Principle in Quantum Gravity STEVEN WEINSTEIN . . . . . . . . . . . . . . . . . . . . .

875

xliv

Continuous Consistent Histories and the Thnneling Time Problem N. YAMADA

878

....................... .

Canonical Quantum Gravity Chairperson: Hideo Kodama Canonical Quantum Gravity Session: Summary Report HIDEO KODAMA

. . . . . . • . . . . • .

882

. . . •

The Equivalence Between the Connection and the Loop Representation of Quantum Gravity R. DE PIETRI . . . . . . . . . . . . . . . . . . . . . .

892

On Generalized Dynamics of Bianchi IX Cosmology D. MLADENOV, V. PERVUSHIN

895

S. GOGILIDZE, A. KHVEDELIDZE,

Canonical Structure of Locally Homogeneous Systems on Compact 3-manifolds HID EO KODAMA

•..

. . . . . . . .

. . . . . . . . . . . .

. . .

.

898

CP-Symmetry in Chiral Gravity ECKEHARD

W.

MIELKE, ALFREDO MACIAS, YUVAL NE'EMAN

901

Problems of Minisuperspace Canonical Quantization A.A. MINZONI, MARCOS ROSENBAUM, MICHAEL P. RYAN, JR.

904

Constants of the Motion and the Quantum Modular Group in (2+1)-Dimensional Gravity V. MONCRIEF, J.E. NELSON

......•........

907

From Reissner-Nordstrom Quantum States to Charged Black Holes Mass Evaporation 911

P.V. MONIZ

The Role of Dilations in Diffeomorphism Covariant Algebraic Quantum Field Theory M. RAINER . . . . . . . . . . • . . . . . . . . . . . . . • . • . . .

914

Loop Quantum Gravity C. ROVELLI . . . . .

917

Eisenhart's Principle Minisuperspace

on the Road to Desingularize Geodesic Motion in . • .

918

MASAYUKI TANIMOTO, TATSUHIKO KOIKE, AKIO HOSOYA

921

MAREK SZYDLOWSKI, MAREK BEISIADA . . . . .

Hamiltonians for Compact Homogeneous Universes

Strings Chairperson: E. Rabinovici Type I Vacua in the Web of String Dualities MASSIMO BIANCHI

. . . . . . . . . . .

925

xlv

2D Induced Gravity from a WZNW System M. BLAGOJEVIC, D.S. POPOVIC, B. SAZDOVIC

928

Dirac Constraint Quantization of Matter-Coupled 2D Dilaton Gravity Models J. CRUZ, J.M. IZQUIERDO, D.J. NAVARRO, J. NAVARRO-SALAS . . . . . . .

931

Exact Path Integral Quantization of 2-D Dilaton Gravity W. KUMMER, H. LIEBL, D.V. VASSILEVICH . . . . . .

936

Minimum Size from Strings and Quantum Cosmology JosE M. RAYA, LUIS J. GARAY, PEDRO F. GONZALEZ-DiAZ, GUILLERMO A. MEN A MARUGAN . . . . . . . . . . . . . . . . . . . . . . . . . .

939

The Casimir Effect Chairperson: V. Mostepanko Casimir Effect in Schwarzschild Geometry KARSTEN BORMANN . . . . . . . . .

943

The Casimiro Experiment: Dynamical Detection of Casimir Forces G. BRESSI, G. CARUGNO, A. GALVANI, R. ONOFRIO, G. Ruoso. .

946

High Sensitivity Measurement of the Casimir Force and Observability of Finite Temperature Effects R. COWSIK, B.P. DAS, N. KRISHNAN, G. RAJALAKSHMI, D. SURESH, C.S. UNNIKRISHNAN. . . . . . . . . . . . . . . . . . . . . . . . . . . ,

949

Pair of Accelerated Frames: A Perfect Interferometer ULRICH H. GERLACH . . . . . . . . . . . . . .

952

Is Sonoluminescence Due to Dynamical Casimir Effect? C.S. UNNIKRISHNAN . . . . . . . . . . . . . . . .

955

Black Hole Thermodynamics Chairperson: T. Jacobson Black Hole Thermodynamics Today T. JACOBSON . . . . . . . . . .

959

Constraints on the Geometries of Black Holes in Classical and Semiclassical Gravity PAUL R. ANDERSON, COURTNEY D. MULL

968

Rindler Space Entropy J.R. ARENAS, J.M. TEJEIRO

971

Black Holes of Constant Curvature MAXIMO BAN ADOS . . . . . . .

974

The 'Ups' and 'Downs' of a Spinning Black Hole C.M. CHAMBERS, W.A. HISCOCK, B.E. TAYLOR

977

xlvi Thermodynamics of Nonsingular Spherically Symmetric Black Hole I. DYMNIKOVA . . . . . . . . . . . . . . . . . . .

980

Black Hole Entropy and Entanglement Thermodynamics HIDEO KODAMA, SHINJI MUKOHYAMA, MASAFUMI SERIU

983

Euclidean Instantons and Hawking Radiation S. MASSA:R, R. PARENTANI . . . . . . . .

986

Toward the New Gravitational Noncommutative Mechanics and Statistical Mechanics of Quantum Black Holes P.O. MAZUR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

989

Loop Corrections for 2D Hawking Radiation A. MIKOVIC, V. RADOVANOVIC . . . . . .

992

Black Hole Entropy from Loop Quantum Gravity C. ROVELLI . . . . . . . . . . . . . . . .

995

The Black Hole Entropy: A Spacetime Foam Approach FABIO SCARDIGLI . . . . . . . . . . . . . . . . .

996

Covariant Path Integrals and Black Holes F. VENDRELL, M.E. ORTIZ . . . . . . .

1002

Quantum Cosmology Chairperson: A. Starobinski Quantum Analysis of the Compactification Process in the Multidimensional Einstein-Yang-Mills System O. BERTOLAMI, P.D. FONSECA, P.V. MONIZ . . . . . . . . . . . . . .

. . 1006

Can Spontaneous Supersymmetry Breaking in a Quantum Universe Induce the Emergence of Classical Spacetimes? O. BERTOLAMI, P.V. MONIZ . . . . . . . . . . . . . . . . . . . . . . . .

1009

The Old Frequency Decomposition Problem in the Light of New Quantization Methods FRANZ EM BACHER . . . . . . . . . . . .

1012

New Results in One-Loop Quantum Cosmology G. ESPOSITO, A.Yu. KAMENSHCHIK . . . . .

1015

Complex Inflaton Field in Quantum Cosmology A.Yu. KAMENSHCHIK, I.M. KHALATNIKOV, A.V. TOPORENSKY

1018

Quantum Inhomogeneous Mixmaster Model and the Origin of the Early Universe A.A. KIRILLOV . . . . . . . . . . . . . . . . . . . . .

1021

Supersymmetric Quantum Cosmology: The Lorentz Constraint ALFREDO MACIAS, ECKEHARD W. MIELKE, JOSE SOCORRO .

1023

xlvii

Can We Obtain Conserved Currents in Supersymmetric Quantum Cosmology? P.V. MONIZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1026

Quantum Cosmology in Some Scalar-Tensor Theories L.O. PIMENTEL, C. MORA . . . . . . . . . . . .

1029

Gravitational Wave Detectors- Bars Chairperson: G. Pizzella Gravitational Wave Resonant Detectors: Status and Perspectives G. PIZZELLA . . . . . . . . . . . . . . . . . . . .

1033

Operation of the Gravitational Wave Detector NAUTILUS P. As TONE et al. . . . . . . . . . . . . . . . . . .

1039

Advances and Perspectives in Capacitive Transducers and Associated Readout M. BASSAN, Y. MINENKOV, V. FAFONE, P. BONIFAZI, M. VISCO, P. CARELLI, M.G. CASTELLANO . . . . . . . . . . . . . . . . . . . . . . . . . .

1042

Transducer Development for the Allegro Detector W.O. HAMILTON, W.W. JOHNSON, E. MAUCELI, M. McHUGH, A. MORSE

.

1045

Calibration and Filtering of Data from a Resonant Mass Gravitational Wave Antenna I.S. HENG, M.E. TOBAR, D.G. BLAIR, E.N. IVANOV . . . . . . . . . . .

1048

Sapphire Dielectric Resonant-Mass Transducer for Gravitational Wave Detection C.R. LOCKE, M.E. TOBAR, E.N. IVANOV, D.G. BLAIR . . . . . . . . . . . .

1053

Gravitational Wave Detectors - Interferometers Chairperson: Robert Spero Gravitational Waves Detection by Phase Interference Y. BEN-ARYEH . . . . . . . . . . . . . . . . .

1057

Sapphire Test Masses and Nb Flexure Suspensions to Reduce the Thermal Noise in Laser Interferometer Gravitational Wave Detectors L. Ju, M. TANIWAKI, D.G. BLAIR, F. BENABID, M. NOTCUTT . . . . .

1060

Design of Suspension System for GEO 600 M.V. PLISSI, K.A. STRAIN, N.A. ROBERTSON, J. HOUGH, C.1. TORRIE, S. ROWAN, S. TWYFORD, S. KILLBOURN . . . . . . . . . . . . . .

1063

Progress on Low-Frequency Active Vibration Isolation S.J. RICHMAN, J.A. GIAIME, R.T. STEBBINS, P.L. BENDER, J.E. FALLER.

1066

Performance of Vibration Isolation System for TAMA300 R. TAKAHASHI . . . . . . . . . . . . . . . . . . .

1069

Ultra-Low Frequency Pre-Isolation J. WINTERFLOOD, D.G. BLAIR . .

1071

xlviii

The Development of an 8m Suspended Power Recycling Interferometer C. ZHAO, M. NOTCUTT, L. Ju, D. BLAIR. . . . . . . . . . . . . .

1074

Gravitational Wave Detectors - Sources Chairperson: Luc Blanchet Gravitational-Wave Sources Luc BLANCHET. . . . . .

1078

Space-Based Gravitational Wave Detectors and the Galactic White Dwarf Binary Background M. BENACQUISTA . . . . . . . . . . . . . . . . . . . . . . . . .

1086

On the Perturbation of Non-Rotating Compact Objects Excited by Massive Sources A. BORRELLI, V. FERRARI, L. GUALTIERI

1089

Calculating the Cosmic Background of Supernova-Generated Gravitational Radiation R.R. BURMAN, D.G. BLAIR, S.J. WOODINGS . . . . . . . . . . . .

1092

Searches of Inspiraling Compact Binaries with Interferometric Detectors SERGE DROZ, ERIC POISSON . . . . . . . . . . . . . . . . . . .

1095

Radiation Backreaction in Spinning Binaries LASZLO A, GERGELY, ZOLTAN PERJES, MATYAS VASUTH

..... .

1098

Cylindrical Collapse and the Gravitational Radiation from van Stockum Space-Time Objects REUVEN OPHER, ERELLA OPHER . . . . . . . . . . . . . . . . .

1101

Black Hole Perturbation Approach to Gravitational Radiation from Inspiralling Binaries MISAO SASAKI . . . . . . . . . . . . . . . . . . . . . . . . . .

1104

Radiation Reaction in Free Fall from Perturbative Geodesic Equations in Spherical Coordinates ALESSANDRO D.A.M. SPALLICCI . . . . . . . . . . . . . . . . . .

1107

Spinning Particle around Black Hole and Gravitational Wave SHINGO SUZUKI, KEI-ICHI MAEDA. . . . . . . . . . .

1111

Supernovae as a Strong Source of Gravitational Radiation A.F. ZAKHAROV . . . . . . . . . . . . . . . . . .

1114

Gravitational Wave Detectors- Future Chairperson: M. Fujimoto Response of a Space born Gravitational Wave Antenna to Solar Oscillations A.G. POLNAREV, G. GIAMPIERI, I.W. ROXBURGH, S. VORONTSOV, K. MARTCHENKOV . . . . . . . . . . . . . . . . . . . . . . . . .

1118

xlix

Observing Coalescing Binaries with Space-Borne Laser Interferometric Gravitational Wave Detectors: Angular Resolution and Astrophysical Parameter Measurements A. VECCHIO, C. CUTLER.

. . . . . . . . . . . . . . . . . . . . . . . . .

1121

General Relativity in Space Chairperson: Kenneth Nordtvedt General Relativity in Space KENNETH NORDTVEDT

1125

D. BARDAS

Development of the Gravity Probe B Payload et al. . . . . . . . . . . . . .

1135

The Technology Heritage of the Relativity Mission, Gravity Probe B SAPS BUCHMAN et al. . . . . . . . . . . . . . . . . . . .

1139

What Can LLR Provide to Relativity? J. MULLER, M. SCHNEIDER, K. NORDTVEDT, D. VOKROUHLICKY

1151

A Robust Squid System for Space Use B. MUHLFELDER, J.M. LOCKHART, M. Luo, T. MCGINNIS

1154

The Clock{s)-to-the-Sun Mission: A New Look 1160

KENNETH NORDTVEDT, ROBERT VESSOT

Testing for Preferred Inertial Frames with Laser Ranging 1163

DAVID VOKROUHLICKY, KENNETH NORDTVEDT

Experimental Tests Chairperson: C.M. Will Session on Experimental Tests . .

C.M. WILL . . . . . . . . .

1167

SEE Project for Measuring the Gravitational Interaction Parameters A.D. ALEXEEV, K.A. BRONNIKOV, N.I. KOLOSNITSYN, M.Yu. KONSTANTINOV, V.N. MELNIKOV, ALVIN J. SANDERS.

. . . . . . . . . . . . . . . . . .

1171

A New Inverse Square Law Test A. ARNSEK, A. CADEZ.

. . . . . .

1174

Influence of the Gravitational Field on the Beam Splitting Process in Atom Interferometry CH.J. BORDE, C. LAMMERZAHL. . . . . . . . . . . . . . . . . . . .

1178

The TIFR Equivalence Principle Experiment R. COWSIK, N. KRISHNAN, C.S. UNNIKRISHNAN .

1181

Local Reference Systems with PPN Parameters S.A. KLIONER, M. SOFFEL . . . . . . . . . .

1184

Quantum Tests of Einstein's Equivalence Principle C. LAMMERZAHL . . . . . . . . . . . . . . .

1187

Techniques for a High-Precision Frame-Dragging Measurement Using an Unsupported Gyroscope in a Drag-Free Satellite BENJAMIN LANGE . . . . . . . . . . . . . . . . . . . . . . .

1190

Preparation for a Determination of G Using a Cryogenic Torsion Pendulum R.D. NEWMAN, M.K. BANTEL . . . . . . . . . . . . . . . . . . . .

1191

A New Experiment to Measure G by Means of a Beam Balance F. NOLTING, J. SCHURR, W. KUNDIG . . . . . . . . . . .

1194

Search for Axions Using a Superconducting Differential Angular Accelerometer H.J. PAIK, E.R. CANAVAN, M.V. MOODY . . . . . . . . . . . . . . . .

1197

Determination of the Gravitational Constant G Using a Fabry-Perot Pendulum Resonator A. SCHUMACHER, H. SCHUTT, H. W ALESCH, H. MEYER . . . .

1200

Possibility of Observing the Gravitational Analogue of the Scalar Aharanov-Bohm Effect in Atom Interferometers C.S. UNNIKRISHNAN. . . . . . . . . . . . . . . . . . . .

1203

Equivalence Principle in Space Chairperson: Y. Jafry Tests of the Equivalence Principle in Space Y. JAFRY . . . . . . . . . . . . . .

1207

MiniSTEP SQUID Position Sensor Development O.H. CLAVIER, R.H. TORII, D.B. DEBRA. . . .

1213

The Using Two-Frequency Lasers in the Gravitational Experiments V.l. DENISOV, N.V. KRAVTSOV, V.B. PINCHUK . . . . . . . .

1216

Test of the Weak Equivalence Principle in Stratospheric Free Fall V. IAFOLLA, E.C. LORENZINI, V. MILYUKOV . . . . . . . . . . . . . . .

1218

Separating Out the Radial-Offset Error in a Two-Sphere Equivalence-Principle Experiment in a Satellite BENJAMIN LANGE. . . . . . . . . . . . . . . . . . . . . . . . . . .

1221

Suppressing Radial Non-Observability in a Concentric Two-Sphere EquivalencePrinciple Experiment by Gravity-Gradient Cancellation BENJAMIN LANGE. . . . . . . . . . . . . . . . . . . . . . . . . .

1224

The Two-Color Transcollimator, a Precision Position Detector for a Satellite Two-Sphere Equivalence-Principle Experiment BENJAMIN LANGE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 Cosmic Tests for a More Explicit Equivalence Principle R.A. VERA . . . . . . . . . . . . . . . . . . . .

1229

Ii

High Sensitive SQUID Based Position Detectors for Application in Gravitational Experiments W. VODEL, H. KocH, S. NIETZSCHE, J. V. ZAMECK GL., R. NEUBERT, H. DITTUS, S. LOCHMANN,

C.

MEHLS

. .

. . .

. . . . . . . . .

. . .

. . .

1232

Observational Cosmology Chairperson: J. Einasto Large Scale Structure of the Universe J. EINASTO

Introduction

1236

................. .

Gravitational Potential Perturbations and Large Scale Bias M. DEMIANSKI, A. DOROSHKEVICH

1240

.......... .

Has the Universe a Honeycomb Structure? J. EINASTO

1243

............ .

Renormalization Group Approach to Einstein Equation in Cosmology OSAMU IGUCHI, AKIO HOSOYA, TATSUHIKO KOIKE

.......

.

Testing Relativistic Zel'dovich Approximation in Spherically Symmetric Model M. MORITA, K. NAKAMURA, M. KASAl

1246

1249

On the Power Law Dependence of the Average Density of the Galaxies in the Universe

C.

SIGISMONDI,

S.

FILIPPI, R. RUFFINI

..............

1252

Power Spectra and Large Scale Structure Formation in Mixed Dark Matter Models with a Cosmological Constant R. VALDARNINI, T. KAHNIASHVILI, B. NOVOSYADLYJ . . . . . . . . . .

1255

Large-Scale Mass Power Spectrum from Peculiar Velocities I. ZEHAVI . . . . . . . . . . . . . . . . . . . . .

1258

Inflation Chairperson: Misao Sasaki Tensor CMB Anisotropy in Open Inflation Models

1262

MISAO SASAKI, TAKAHIRO TANAKA, YOSHIHIRO YAKUSHIGE

Canonical Perturbations in the One Bubble Universe JAUME GARRIGA, XAVIER MONTES, MISAO SASAKI, TAKAHIRO TANAKA

1274

On Classical Anisotropies in Models of Open Inflation JAUME GARRIGA, V.F. MUKHANOV .

. . . .

. .

1277

.

Second-Order Reconstruction of Inflationary Dynamics Compatible with Recent COBE Data ECKEHARD

W.

MIELKE, ALFREDO MACIAS, ALBERTO GARCIA

. .

. . . . ..

1280

Iii

Cosmological Vacuum in Unified Theories V.N. PERVUSHIN, V.1. SMIRICHINSKI .

..................

1283

Topological Defects Chairperson: R. Durrer Can a Self-Gravitating Thin Cosmic String Obey the Nambu-Goto Dynamics? B. BOISSEAU, C. CHARMOUSIS, B. LINET. . . . . . . . . . . . . . . . . . . 1287 Current-Carrying Cosmic String Loops Leading to Vortons ALEJANDRO GANGUI, EDGARD GUN ZIG . . . . . . . .

. .

1290

On the Formation of Hierarchical Astronomical Objects in the Cosmic String Scheme TETSUYA HARA, SIGERU MIYOSHI, PETRI MAHONEN · . .

1293

On the Perturbation of Domain Wall Coupled to Gravitational Waves AKIHIRO ISHIBASHI, HIDEKI ISHIHARA . . . . . . . . . . . . . . . . . 1296 Formation Probability of Non-Topological Electroweak Strings M. NAGASAWA, J. YOKOYAMA . . . . . . . . . . . . . .

· . .

1299

..

1302

Testing Theoretical Predictions of CMB Anisotropies Induced by Scaling Seeds Against Current Observational Data M. SAKELLARIADOU, R. DURRER, M. KUNZ, C. LINEWEAVER . . . . . . . . .

1305

Numerical Experiments in Cosmic String Networks in Terms of a Three-Scale Model M. SAKELLARIADOU, M. HINDMARSH, G.R. VINCENT

1308

Cylindrical Domain Walls and Gravitational Waves - Einstein Rosen Wave Emission from Momentarily Static Initial Configuration KOUJI NAKAMURA, HIDEKI ISHIHARA . . . . . . . . . . . . . . . . .

Topology of the Universe and Topological Defects JEAN-PHILIPPE UZAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311

Early Universe Chairperson: N. Deruelle FRW-Geometry Evolution due to Quantum Effects FRANK ANTONSEN, KARSTEN BORMANN . . . . .

· . .

1315

Exact Solutions and Growth of Cosmological Perturbations in the 'fransition Era from the Radiation to Matter Dominance ANDRES ARAGONESES, VICENQ MENDEZ, DIEGO PAVON, WINFRIED ZIMDAHL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318 Higher Derivative Theory with Viscosity A. BEESHAM, B.C. PAUL, S. MUKHERJEE

. . . . . . . . . . . . . . . . . . 1321

liii

Baryogenesis Motivated on String CPT Violation . . . . . . . . .

O. BERTOLAMI . . . . . . . . . . . . . . .

1324

Zero Energy Modes of Massless Fermions in S3 x R Spacetime M.A. DARIESCU, C. DARIESCU

. . . . . . . . . . . . . .

. . . . . 1327

Self-Consistent Solutions for Low-Frequency Gravitational Background Radiation 1330

G. DAUTCOURT. . . . . . . . . . . . . . . . . . . . . . . . .

Cosmological Structures in Generalized Gravity . . . . . . . . . . . . . . . .

1333

. . . . . . . . . . . . . . . .

1336

. . . . .

1339

DIEGO PAVON, VICENC MENDEZ, JosE M. SALIM. . . . . . . . . . . . . . .

1342

J. HWANG

. . . . . . . . . . . . .

Bubble Dynamics at Finite Temperature CHUL H. LEE. . . . . . . . .

Energy in the Primordial Plasma MERAV OPHER, REUVEN OPHER

The Cosmic Fluid is Essentially Transparent to Gravitational Radiation Origin of Structure in a Supersymmetric Quantum Universe P. VARGAS MONIZ

1345

. . . . . . . . . . . . . . . . . .

Energy Conditions and Galaxy Formation MATT VISSER

1348

........... .

Cosmological Perturbations and Large-Scale Conservation Quantities W. ZIMDAHL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1351

Cosmic Microwave Background Chairperson: Naoshi Sugiyama A Quick View of the CMB Session and Thermal History & CMB Anisotropies . .

1355

...

1367

...

1370

. . . . . . . . . . . . . .

1373

. . . . . . . . . . . . . . .

1376

A. WOSZCZYNA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1379

NAOSHI SUGIYAMA

. . . . . . . . . . . . . . . . . . . . . . .

The Signature of the Negative Curvature of the Universe in CMB Maps V.G. GURZADYAN, S. TORRES

. . . . . . . . . . . . . . . . ..

Recent Results of the Tenerife CMB Experiments G.M. GUTIERREZ, R. REBOLO, R.J. HOYLAND, R.D. DAVIES, R.A. WATSON, A.N. LASENBY, A.W. JONES, S. HANCOCK . . . . . . . . . . . . .

CMB Anisotropies Caused By Gravitational Waves: A Parameter Study T. KAHNIASHVILI, R. DURRER

. . . . .

Can the CMBR Dipole be Cosmological? DAVID LANGLOIS

. . . . . . . . . . .

On Ergodic Perturbations in the Open Universe

liv

N onsingular Cosmology Chairperson: Mario Novello The Program of the Eternal Universe MARIO NOVELLO . . . . . . . . .

.

1383

The Causal Interpretation of Dust and Radiation Fluids Non-Singular Quantum Cosmologies J. ACACIO DE BARROS, M.A. SAGIORO LEAL, M. PINTO-NETO . . . . . . . . .

1403

Inflationary Cosmology in Weyl Integrable Geometry J.C. FABRIS, J.M. SALIM, S.L. SAUTU . . . . . . . . . . .

........

1406

Globally Hyperbolic Geodesically Complete Cosmological Model LEONARDO FERNANDEZ JAMBRINA . . . . . . . . . . . . . . . . . . . . .

1409

Relativistic Cosmological Hydrodynamics J. HWANG, H. NOH . . . . . . . . . .

. . . .

1412

An Alternative Perspective in Quantum Mechanics and General Relativity B.G. SID HARTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1415

Fluctuational Cosmology B.G. SIDHARTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1418

Dark Matter Chairperson: R. Cowsik Dark Matter - An Introduction R. COWSIK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1422

Baryonic Dark Matter FRANCESCO DE P AOLIS, PHILIPPE JETZER, GABRIELE INGROSSO, MARCO RONCADELLI . . . . . . . . . . . . . . . . . . . . . . . . . . .

1426

Microlensing Implications for Halo Dark Matter FRANCESCO DE P AOLIS, PHILIPPE JETZER, GABRIELE INGROSSO, MARCO RONCADELLI . . . . . . . . . . . . . . . . . . .

1429

Landau Damping of Fermion Perturbations in an Expanding Universe

S. FILIPPI, C. SIGISMONDI, L.A. SANCHEZ

. . . . . . . . . . . .

. .

1432

Some Geometrical Models the Possible Class of Particles Able to Travel Faster than Light M.Yu. KONSTANTINOV . . . . . . . . . . . . . . . . . . . . . . . . . .

1435

Evolution of Power Spectrum for Inos H.W. LEE . . . . . . . . . . . .

. . . . . . . . . . .

1438

Accelerator Test of the Dark Matter Neutrino Hypothesis HISAKAZU MINAKATA . . . . . . . . . . . . . . . . . . . . . . . . . . .

1441

Iv

The Oscillating G Model: A Possible Explanation for the Apparent Periodicity in the Galaxy Distribution and for the Cosmological Non-Baryonic Matter HERNANDO QUEVEDO, MARCELO SALGADO, DANIEL SUDARSKY

. • . . . . .

.

1444

Boson Halo: Scalar Field Model for Dark Halos of Galaxies FRANZ

E.

SCHUNCK .

. . . . . . . . .

. . . .

. . .

. . . 1447

.

Experimental Evidence of a New Type of Quantized Matter with Quanta as Integer Multiples of the Planck Mass .,.

1450

. . . .

. . • . . . . . . . . . .

1454

. . . . . . .

. . . . . . . . . . . . .

1458

KLAUS VOLKAMER, CHRISTOPH STREICHER

. . . .

. . . . . . . ..

A Theory of Superluminal Particles with Real Mass Content in Agreement with Experimental Evidence of a New Type of Quantized Matter KLAUS VOLKAMER, CHRISTOPH STREICHER

Microlensing by Noncompact Objects A.F. ZAKHAROV, M.V. SAZHIN

Astro-particles Chairperson: A. Dar Primordial Nucleosynthesis with Neutrino Degeneracy and Gravitational Constant Variation HYUN Kyu LEE

1462

....................... .

The Search for Antimatter in the Universe ALDO MORSELLI

. . . . .

1465

. . . . . .

Neutrino Transport in Type II Supernovae Boltzmann Solver vs. Monte Carlo Method . . 1468

SHOICHI YAMADA, HAN-THOMAS JANKA, HIDEYUKI SUZUKI

Gravitational Lenses Chairperson: A. O. Petters Some Global Results on Gravitational Lensing . . . . .

1472

....

1485

. . . .

1488

On the Classification of Stable Caustics W. HASSE, V. PERLICK, M. KRIELE . . . . . . . . . . . . . . . . . . . . .

1491

A.O. PETTERS

............. .

Defocusing Microlensing R. DE RITIS, V.1. MAN'KO, A.A. MARINO, G. MARMO,

S. CAPOZZIELLO, A. SIMONI

. . . . . . . .

. . . . . . . . .

. . . .

. • • . .,

The Fermat Principle in General Relativity and the Gravitational Lensing Effect FABIO GIANNONI, ANTONIO MASIELLO, PAOLO PICCIONE

Ivi

Properties of Point Mass Lenses on a Regular Polygon and the Problem of Maximum Number of Images S. MAO, A.O. PETTERS, H.J. WITT . . • . . . . . . . . • • . . . .

. . . .

1494

Counting Formulas and Bounds on Number of Fixed Points Due to Point Mass Lenses A.O. PETTERS, F.J. WICKLIN . . . . • . . . . . . . . . . . . . .

1497

Gravitational Lens Equation Near Cusps . • . . . . . A.F. ZAKHAROV

1500

.

Extragalactic Sources Chairperson: A. 1'reves Overwhelming Evidence for a Massive Black Hole in a Nearby Galaxy from Maser Emission Observations EYAL MAOZ

. . . 1504

....... .

Astrophysical Black Holes Chairperson: M. Abramowicz Some Effects due to Interaction of Charged Particles and Charged Rotating Black Holes . . . 1508

Z. STUCHLfK

QPO's Chairperson: L. Stella Lense-Thirring Precession and QPOs in Low Mass X-Ray Binaries L. STELLA, M. VIETRI . .

. .

. . . .

. . . . . . . . . .

. .

1512

General Relativity and Quasi-Periodic Oscillations PHILIP KAARET

. . • . . . . . . • .

1516

Binary Neutron Stars Chairperson: Takashi N akam'Ura Report of the Workshop on Binary Neutron Stars/Black Holes T AKASHI NAKAMURA

. . .

. . .

. .

. . . . . .

1520

Dynamics of Relativistic Superfluid in Neutron Stars . . . . . . .

1524

Relativistic Studies of Close Neutron Star Binaries G.J. MATHEws, P. MARRONETTI, J.R. WILSON . .

1527

DAVID LANGLOIS

. . . . . .

. .

Gravitational Waves from Coalescing Black Hole MACHO Binaries TAKAS HI NAKAMURA, MISAO SASAKI, TAKAHIRO TANAKA KIP S. THORNE

1530

Ivii

Mayan Eclipsing Binary Radiopulsar Be a Strong-Field Gravitational Laboratory?

1533

IGOR!. SALUKVADZE

Location of the Innermost Stable Circular Orbit of Binary Neutron Stars in the First Post Newtonian Approximation MASARU SHIBATA.

. . . . . . . . .

. . . .

. . . . . . . . . . . . .

1536

Relativistic Equations of Motion for Binary Systems with Monopole, Spin and Complete Quadrupole Moments

C. Xu, Y.

HE,

X. Wu . . . . . . . . . . . . . . . . . . . . . . . . .

1539

Microlenses Chairperson: Philippe Jetzer Gravitational Microlensing Research

1543

PHILIPPE JETZER . . . . . . . .

Microlensing Results and the Galactic Models FRANCESCO DE P AOLIS, PHILIPPE JETZER, GABRIELE INGROSSO

1561

MACHO versus HST: How Bright Can Dark Matter Be? . . . .

1564

Noncompact Astronomical Bodies as Microlenses M.V. SAZHIN . . . . . . . .

1567

E.J. KERINS

• . .

. . . . . . . . .

A.F. ZAKHAROV,

Gamma Ray Bursts Chairperson: David L. Band The Gamma-Ray Burst Mystery DAVID L. BAND. .

. .

1571

. . . .

Can Internal Shocks Produce the Variability in GRBs? SHIHO KOBAYASHI, TSVI PIRAN, RE'EM SARI.

1591

.

Gamma-Ray Bursts from Neutron Star Binaries

1594

G.J. MATHEWS, J.R. WILSON, J. SALMONSON

A Unified Model for Gamma Ray Bursters, Galactic Microquasars and Ultrahigh Energy Cosmic Rays ERELL A OPHER, REUVEN OPHER . . .

. . . .

. . . . . . . . . .

1597

Spectral Implications of Variability in GRB Fireballs RAVI P. PILLA, ABRAHAM LOEB

...... .

1600

Numerical Simulations of Colliding Neutron Stars M. RUPPERT, H.-TH. JANKA . . . . . . . . .

1603

Iviii

Boson Stars Chairperson: Eckehard W. Mielke

Boson Stars: Early History and Recent Prospects ECKEHARD W. MIELKE, FRANZ E. SCHUNCK . .

. . . 1607

Localized Mass and Spin in 2+1 Dimensional Topologically Massive Gravity A. EDERY, M.B. PARANJAPE . . . . . . . . . . . . . . . . . . . . . . . . 1627 On the Stability of Real Scalar Boson Stars PHILIPPE JETZER, DAVID SCIALOM

. . . . . . . . . . . . . . . . . . . . .

1630

Differentially Rotating Boson Stars FRANZ E. SCHUNCK, ECKEHARD W. MIELKE

1633

List of Participants

1636

Author Index . . .

1646

PLENARY SESSIONS

2

DISCRETE GRAVITY TULLIO REGGE Dept. of Physics, Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129, Torino, Italy

I have always been a great admirer of David Hilbert and of course I am not alone. A delightful book is his "Anschauliche Geometrie" written in collaboration with Vossen-Cohn. It it not a standard textbook or a manual nor is it a work where the authors popularize geometry. Clearly it has been written for colleagues who want something different and stimulating. I was always intrigued by the splendid set of pictures illustrating the regular tesseract in 4 dimensions, the analogs of platonic solid. In the book they appear as kind of skeletons built out of sticks or bones which represent the edges of the cells which themselves are regular polyhedra. As you travel inside the tesseract you move smoothly from cell to cell and you can glue adjacent cells into a subset where geometry is euclidean as long as the subset does not include in its interior a bone. The same happens when you try to build a polyhedron out of paper, usually you start from a tree of faces which can be flattened on a plane but as soon as you glue faces around a vertex you go out of the plane and get intrinsic curvature. In the two cases thus considered of dimension d = 2,3 curvature is a distribution with support on a skeleton of co dimension 2. The complement of the skeleton has usually a non trivial first homotopy group 11"1 . Parallel transportation along a closed loop enclosing a bone will rotate a vector along the bone as axis by an angle c which is easily evaluated by elementary methods and which depends only on the homotopy class of the loop. On a polyhedron the bone reduces to a vertex V and c , often called deficit angle, is given by the sum :

(1) where i runs on all faces sharing the same vertex V and Cti is the internal angle of the face hinging on V . Indeed if c = 0 there is no curvature left on V . We have then the discrete version of Gauss-Bonnet curvatura integra theorem: LE(V) = 411"(1- g)

(2)

v which can be easily deduced from Eulers theorem, 9 is here the genus of the polyhedron. Variation of the above equation leads to:

2:/Scv v

=0

(3)

which we consider later in a different context. To a generic element 9 E 11"1 we associate similarly an element A(g) of the appropriate orthogonal or pseudoorthogonal rotation group depending on the flat metric adopted. For the d = 4 dimensions of general relativity 1 this group will be of course the Lorentz group 80(3,1) and the

3

deficit angles may be hyperbolic. We may then say that the set of bones equipped with deficit angles is all what is left of the Riemann tensor and that the the representation 71"1 A 80(3,1) yields the holonomy group. In codimension 3 we find objects, called nodes, on which hinges a subset of bones. Nodes do not exists on a polyhedron but are quite visible as vertices in the pictures in Hilbert's book. Elements in 71"1 bones sharing the same same node are not independent since there is a discrete Bianchi identity of the kind: 9192 ... 9m

=1

and A (91) A (92) ... A (9m)

= Identity Matrix

(4)

where the ordering of the 9k, k = 1, ... , m is crucial but need not to be discussed here. This discrete geometry would not be so interesting if it did not have an action. Luckily there is a very good candidate for an action 8 given by: 8= ~ c(~)L(~)

(5)

(3

where ~ labels bones, the sum runs on all bones and L(~) is the measure of the bone support in the appropriate euclidean or pseudoeuclidean metric. For d = 2 we have L(m = 1 and the sum reduces to the discrete 2 and therefore to a divergence. For d = 3 we have L(f3) = l(f3) = length of f3. Whatever the dimension d we have:

~ 8c(m L(m =0

(6)

(3

that is variation of 8 can be performed as if the c(~) were constant. In fact 3 is a particular example of this formula. For d = 3 the lengths l(~) are the discrete equivalent of the metric and are all independent, it follows that the variational equations are now c(~) = O. All these results are obviously in agreement with what happens in a differentiable manifold. For d = 4, as expected, the variational equations are more complex but anyway very interesting. J.Cheeger et. al. 3 have investigated in detail the way these discrete equations approach the classical Einstein field equation in the continuous limit and found complete agreement. T.D.Lee has shown that also the inverse limit holds, i.e., if a differentiable manifold gradually sharpens and becomes a tesseract then the Einstein-Hilbert action has 5 as limit. It is possible to build approximate models of a Friedmann universe where the space slice is a regular tesseract and the field equation reduce to 'a finite set of trigonometric equations which can be easily solved. The accuracy of these solutions is remarckable give the crude approximation. However simplicial gravity is not merely an approximation method, in this it would have to compete with very efficient numerical codes which have been used to analyze the collision between black holes. What makes it attractive is the fact that a simplicial manifold is always a manifold at every stage and not merely a discretization of the metric tensor. At the moment it is the only lattice model with a local gauge invariance and the only model known containing gravitons for d = 4.

4

We should not be surprised if there are already in the market several variations on the theme. A most successful one is the so called dynamical triangulation approach (DT) which was originally proposed by Weingarten. In this approach the basic cells have fixed shapes, usually regular poligons or polyhedra but the curvature is induced instead by varying the connectivity of the complex. In this way the computations problems become purely combinatorial and find a natural outlet into statistical mechanics and into the theory of random surfaces rather than in gravity. A different way to look at discrete gravity originates from an apparently unrelated field. Techniques for dealing with large number of angular momenta have found wide application in nuclear and atomic physics and utilize as chief tools the Clebsch-Gordan coefficients and the Racah coefficients. Following the group theoretical work of Wigner on the unitary representations of SU(2) these tools became known as the 3j and 6j symbols respectively. Although they can be evaluated explicitly through well known formulas it took some time before we had efficient asymptotic formulas for the case of very large angular momenta where conventional formulas fail. The Racah coefficient or 6j symbol is the simplest of a hierarchy of so called 3nj symbols which describe the recoupling of an arbitrary set of angular momenta. Each symbol is identified by a polyhedron with triangular faces and experts have devised since long time recursive formulas which relate classes of symbols of which the Biedenharn-Elliot formula is the prototype. In 4 the asymptotic behaviour of the 6j symbols is derived and the idea that the hierarchy of 3nj symbols form what today is called a topological theory is discussed . A most surprising fact is that if we write a generic 3nj symbol as a sum over products of 6j's and we try to retrieve its asymptotic behaviour from that of the single 6j factors we arrive to a formula which looks like a sum over Feynman histories for a d = 3 gravity theory. If we use then the stationary phase method we we find that most of the contribution to the sum come from configurations of the angular momenta which exist classically in R3 as expected. This startling intrusion of gravity into angular momentum theory is very interesting but poses puzzling problem which are as yet unsolved. The theory obviously describes something happening in R3 but without reference to time. Replacing the SU(2) group with SU(l, 1) yields a structure endowed by pseudoeuclidean metric but at the cost of introducing an extra factor i in the exponent thus obtaining a statistical model which has no place in such a metric. Replacing SU(2) with a generic Lie group in the hope of reaching a similar result in a higher dimensional space leads us to collide with an infamous obstacle. As a rule the product of unitary irreps of Lie groups are not simply reducible and one needs to introduce an extra index into the 3j symbols in order to label this additional degeneracy. The definition of the extra index is conventional and depends on the choice of basis for the irreps. Moreover the topology of the recoupling coefficient still matches that of a 3 dimensional graph with the proviso that the single angular momentum associated with each side is now replaced by a full set of observables labeling the irreps of the group. There is no explicit formula for the recoupling coefficient and their asymptotic treatment is still science fiction. All this implies that maybe there is interesting math Hying ahead but no great hopes for quantum gravity. Later on it was recognized that we may replace SU(2) with its quantum version

5

SUq (2) thus obtaining further examples and very interesting ones of topological theories and in fact it is possible to obtain in this way recently discovered 3-manifold invariants. Quantum groups are clearly associated asymptotically with background spaces of constant curvature related to q . I cannot end this brief review without quoting the work done on the quantization of gravity by H.Hamber and R.Williams 2. These authors have applied the Feynman sum over histories to graphs of large size and found evidence for a phase transition between a polymer-like phase and one which resembles the world we live in. Immirzi has discussed at length the relationship between discrete gravity and Ashtekar variables. I am sure that I have omitted some important work but I am in the rather embarassing position of having been an outsider in this field during the past 30 years. The enclosed bibliography is neither complete nor balanced and it is intended only as a starting point for navigation into discrete gravity. References

T.Regge, Nuovo Cimento 10, 558 (1961) H.W. Hamber,R.M. Williams, Nucl. Phys. B 487, 345 (1997) J. Cheeger, W. Muller and R. Schrader ,Comm. Math. Phys. 92,405 (1984) G. Ponzano and T. Regge, in Spectroscopic and Group Theoretical Methods in Physics, (North Holland, Amsterdam 1968). 5. S. Mizoguchi and T. Tada, Phys. Rev. Lett. 68, 12 (1991)

1. 2. 3. 4.

6

THE QUANTUM AND GRAVITY: THE WHEELER-DEWITT EQUATION BRYCE DEWITT. Center for Relativity, Department of Physics, The University of Texas at Austin Austin, TX 78712-1081, USA Abstract This equation should be confined to the dustbin of history for the following reasons: 1) By focussing on time slices it violates the very spirit of relativity. 2) Scores of man-years have been wasted by researchers trying to extract from it a natural time parameter. 3) Since good path integral techniques exist for basing Quantum Theory on gauge invariant observables only, it seems a pity to drag in the paraphernalia of constrained Hamiltonian systems. 4) In the case of mini-superspace models, gauge invariant transition amplitudes defined by the path integral do not satisfy any local differential equation; they satisfy the Wheeler-DeWitt equation only approximately.

1

Introduction

When I was first asked to address this Conference no mention was made of a topic, so I thought it might be fun to ride an old hobby hors,e of mine: the topology of the Universe. I have here in my hand an ad from Cambridge University Press plugging one of their latest releases, a book by Peter Coles and George Ellis entitled Is the Universe Open or Closed? The Density of Matter in the Universe - as if the density of matter were the principal determining factor! It is true that a large enough density can force the Universe to be closed, but if the density is less than a critical value nothing can be inferred. It has always been a mystery to me why cosmologists still don't acknowledge this despite the fact that mathematicians have known for over a quarter of a century that there exists an infinity of compact 3-spaces of constant negative curvature, each having the familiar open hyperboloid as its universal covering space and each being associated with a knot in Euclidean 3-space. Over 99 of all known knots have such associated 3-spaces. Our universe could be anyone of them. None would be incompatible with either inflation or the isotropy of the cosmic microwave background. I was starting to prepare an account of these 3-spaces, which incidentally were discovered by the mathematician Thurston, when I received an e-mail message informing me that the Conference organizers were expecting me to talk about the Wheeler-DeWitt equation! Some of you here have heard me refer to it as "that damned equation." To explain why, I first have to reminisce a bit - which is what we old geezers are best at. John Wheeler, the perpetuum mobile of physicists, called me one day in the early sixties. I was then at the University of North Carolina in Chapel Hill, and he told me that he would be at the Raleigh-Durham airport for two hours between planes. He asked if I could meet him there and spend a while talking quantum gravity. John was pestering everyone at the tiIIte with the question: What are the properties of the quantum mechanical state functional W and

7

what is its domain? He had fixed in his mind that the domain must be the space of 3-geometries, and he was seeking a dynamical law for w. I had recently 1

read a paper by Asher Peres which cast Einstein's theory into HamiltonJacobi form, the Hamilton-Jacobi function being a functional of 3-geometries. It was not difficult to follow the path already blazed by Schrodinger and write down a corresponding wave equation. This I showed to Wheeler, as well as an inner product based on the Wronskian for the functional differential wave operator. Wheeler got tremendously excited at this and began to lecture about it on every occasion. I wrote a paper on it in 1965, which didn't get published until 1967 because my Air Force grant was terminated, and the Physical Review in those days was holding up publication of papers whose authors couldn't pay the page charges. My heart wasn't really in it because, using a new kind of bracket discovered by Peierls, I had found that I could completely dispense with the cumbersome paraphernalia of constrained Hamiltonian systems and build a manifestly gauge covariant quantum theory ab initio. But I thought I should at least point out a number of intriguing features of the functional differential equation, to which no one had yet begun to devote much attention: (1) The curious hyperbolic metric that appears in it, which describes a 6-dimensional space having the structure R x ~5 where ~5 is the coset space 8£(3, R) / 80(3) and R is the "timelike" direction. (2) The bizarre logical pathways that one has to follow in interpreting it, which are especially novel when 3-space is compact. (3) The fact that the wave functional is a wave function of the Universe and therefore cannot be understood except within the framework of a many-worlds view of quantum mechanics. The last two features are the most important. In the long run one has no option but to let the formalism provide its own interpretation. And in the process of discovering this interpretation one learns that time and probability are both phenomenological concepts. A few years ago I was at a Workshop in southern Spain and, at its end, one of the participants polled the others on their views regarding the nature of time (since the topic of the conference was the "arrow of time" ). I asserted that time is what a clock measures, nothing more. Enough other participants shared my view, that when the results of the poll were passed to the press several Spanish newspapers printed columns headed "Physicists claim time does not exist!" As I told Tsvi Piran, if the organizers of this conference really wanted me to talk about the Wheeler-DeWitt equation they should be quite aware where I stand on it. It has played a useful role in getting physicists to frame important and fundamental questions, but otherwise I think it is a bad equation, for the following reasons: (1) By focussing on time slices (spacelike 3-geometries) it violates the very spirit of relativity. (2) Scores of man-years have been wasted by researches trying to extract from it a natural time parameter. (3) Since good path-integral techniques exist for basing quantum theory on gauge invariant observables only, it seems a pity to drag in the paraphernalia of constrained Hamiltonian systems. I subscribe 100 to the modern view that the quantum theory should be defined by the path integral. I am going to show you how the path integral can be used both to resolve conceptual issues and to yield gauge invariant transition amplitudes that are operationally well defined. Except in special cases these amplitudes do not satisfy any local differential equation. They

8 satisfy the Wheeler-DeWitt equation only approximately. This means that, generically, the Wheeler-DeWitt equation is wrong, even assuming that the difficult issues of quantum gravity's perturbative nonrenormalizability can be resolved, via string theory or whatever. One may legitimately use the WheelerDeWitt equation, and W K B approximations to its solution, in analyzing such things as the role of quantum fluctuations in the early universe. But it is wrong to use it as a definition of quantum gravity or as a basis for refined and detailed analyses. As I told the conference organizers, decades have passed since I last gave more than a passing glance at the Wheeler-DeWitt equation, and therefore I beg forgiveness of those many persons of whose work I am largely ignorant and will fail to acknowledge. I shall have nothing to say of the important work of the Ashtekar school. I shall also have time only for passing reference to the hoped-for future role of the path integral in the consistent-histories framework for viewing the wave function of the universe, with which I am spiritually in full accord. I hope these lacunae will be filled by others during this conference.

2

Mini-superspace model

I am going to use the simplest of all mini-superspace models to illustrate my ideas: the Friedmann-Robertson-Walker universe carrying a spatially constant scalar field. This model has probably been studied ad nauseum, and I am sure there are those in this audience who will be shaking their heads at my ignorance of and repetition of earlier work. But maybe you will find a few new ideas here. In units for which Ii c 167rG 1, and with the cosmological constant assumed to vanish, the action functional for the model is

= =

=

s=

!

Ldt,

(I)

where

(2) with R{t) and o{t) restricted to have positive values only. Here R is the absolute value of the radius of curvature of the universe, n is the volume of the universe'when R = 1, 0 is the lapse function, cfJ is the scalar field, and V{cfJ) includes any scalar mass or self-coupling terms. The sign in front of the oR term is + or - according as the curvature of 3-space is positive or negative; this term is absent when 3-space is flat. When V = 0 the scalar field is said to be minimally coupled to gravity. The case of conformal coupling adds further cp..dependent terms to L. Dots denote differentiation with respect to t, which is an arbitrary independent variable whose only role is to fix the arrow of time and to order events. The functional S is invariant under the I-dimensional diffeomorphism group, GHere my earlier remarks on the existence of compact negative-curvature 3-spaces become relevant. The mini-superspace model of a universe of infinite volume would have an infinite amount of action and could not be quantized.

9

the infinitesimal actions of which are

=

do

-d(Odt)/dt, }

-mt,

dR

(3)

-~dt, where dt is an arbitrary infinitesimal function of t of compact support. The dynamical equations are

0=

!!=n{6[R(~:f±R]-~R3(~;f-R3V(¢)}'

o =

dS

dR

(4)

2 (R dR ) _ dR ± 1 + ~R2 (d¢)2 _ ~R2V(¢)] (,5) dr dr dr 4 dr 2

= 6no [2~

o where

(6) 'I

is the proper time:

dr = o(t)dt.

(7)

Equation (4) is the constraint on the Cauchy data of the model arising from the diffeomorphism invariance. When V = 0 these equations can be solved in closed form in terms of elementary functions. In this case the Lagrangian L is invariant under the global symmetry operation

¢ -+ ¢

+

(8)

const.,

and by choosing the "const." appropriately one finds, in the positive-curvature case, R

=

dr

Rmax (COSh

~¢)

-1/2

1 ( 1)

2y'3Rmax cosh y'3¢

(9)

-3/2

(10)

d¢,

while in the negative curvature and flat cases one finds R =

( ± sinh ~¢

)-1/2 and R

=

const.

x

'11/3

=

const.

x exp

const.

x

(2Ta¢)

re-

spectively. In the positive-curvature case R has a turning point at Rmax and the universe has a finite lifetime (from big bang to big crunch) given by

1

T = 2y'3Rmax

fOO (

1)

cosh y'3¢

-3/2

d¢ = 1.198 Rmax .

(11)

-00

In the negative-curvature and flat cases R has no turning point but ranges between 0 and 00, going from a big bang or toward a big crunch with a history that covers an infinite amount of time.

10

3

Clock variables

Equation (9), and the corresponding equations in the negative-curvature and flat cases, express correlations between R and cpo Such correlations exist also when V is nonvanishing, and therefore the possibility exists of using either the scalar field or the universe itself as a clock, the values of cp and R being their "readings." cp is perhaps a better clock variable than R, in the classical positive-curvature minimal-coupling case at any rate, because it doesn't turn around. However, in the quantum theory both Rand cp fluctuate, running forward as well as backward as t increases. This is an illustration of the wellknown general rule: No physical clock is perfect. Classically one always has to make sure that the clock is started running forward, not backward, and, in addition to the clock mechanism, one may have to introduce a device to count cycles. Quantum mechanically, one cannot avoid an uncertainty in the clock's reading arising from the finiteness of its mass. Ideal, perfect time can never be represented by an Hermitian operator because it is conjugate to the energy, which always has a spectrum bounded on one side. 4

The path integral

The path integral for the model is I

=

[da][dR][dcp]

N / eiS[Q,R,,plp.[a, R, cpJ, [da][dR][dcp],

(12)

II da(t)dR(t)dcp(t),

(13)

the integration ranges being given by

o< a(t) < 00,

0

< R(t) < 00,

-00

< cp(t) < 00 for

all t,

(14)

N being a normalization constant, and p.[a, R, cp] being the measure functional for the integral, abollt which I shall have a lot to say later. Because of the diffeomorphism invariance of the model and because the lapse function is arbitrary (subject to the constraint (14», the t integration in eq. (1) need not go from -00 to 00. No generality is lost if the limits are taken to be arbitrary finite values L and t+, with L < t+. The formal infinite product (13) is then replaced by

[dR][dcp] [da]

TIL+ at a later instant if they had the values R_ and 1/>_ at an earlier instant. Notice that I have merely said "amplitude," not "probability amplitude." Probability comes later, phenomenologically, after one has decided how to look at the system. Some authors have tried to keep R and I/> on an equal footing throughout the analysis, by regarding both simultaneously as commuting Hilbert space operators and viewing the eigenvector IR- ,1/>-) as representing a state in which R has the value R_ and I/> has the value 1/>_. My friend Chris Isham, in his Salamanca lectures five years ago,2 convincingly showed the untenability of this interpretation, chiefly stemming from the difficulty of normalizing a wave function like (R_ ,1/>-1'11). Fortunately the path integral does not require a commitment to such an interpretation. l.From the point of view of the path integral the chief thing wrong with the interpretation is that neither R nor 1/>, by itself, is diffeomorphism invariant, and hence neither is an observable. What one can do is construct diffeomorphism invariants that try to embody the basic idea behind the interpretation. Consider the integrals

r( 1/>')

(t)5(R(t) - R')R(t)dt.

(18)

t+

L

It is easy to check that they are both invariant under the transformations (3) and therefore are observables. For any history satisfying the boundary conditions (16), r(I/>') can be regarded as the value of R when I/> has the value 1/>', and ') and -+

Here the plus sign attached to the product symbol reflects the fact that p, like (3, is unrestricted at ¢l- and ¢l+. It is easy to check that the classical on-shell equation oS/o(3 = 0 yields a new form for the constraint: (54) while the equations oS/op = 0 and oS/or = 0 yield respectively the second of eqs. (46) and the same dynamical equations as those to which the Lagrangian in the form (26) gives rise.

18

10

Computation of the measure functional

By computing the second functional derivatives of the action (51) with respect to r,p and (3 one easily finds the following form for the Jacobi field operator:

(55)

where A

=

{J2X/{Jr 2 ,

B

=

{J2X/{Jp2,

C

=

{J2 X/{J(32,

D

=

{J2X/{Jr{Jp,

E

=

{J2 X/{Jr{J(3 ,

F

=

{J2 X/{Jp{J(3.

The only matrix element we shall need explicitly is

C=

n- 1r- 31lE(3-3 .

(57)

The advanced and retarded Green's function for the operator (55) are easily verified to be

G+(rjJ,q/)

=

-6(¢,¢'J ( :

0

0

0

0

0

C- 1

0

-1

1

0

-FC- 1

EC- 1

) - 9(¢' - ¢J (

FC-' ) -EC- 1 0

+O(rjJ - rjJ'),

G-(rjJ,rjJ')

=

-6(MJ ( :

(58) 0

0

0

0

0

C- 1

0

-1

1

0

-FC- 1

EC- 1

) +9(¢-¢'J (

FC-' ) -EC- 1

+O(rjJ - rjJ'),

0 (59)

where 8 is the step function. From these expressions the commutator function immediately follows

1

-FC- 1

o

EC- 1

o

-EC- 1

)

+O(rjJ-rjJ'), (60)

which, in turn, yields the equal-time Peierls brackets (r,r)

(r,p)

(p,r)

(P,p)

«(3, r)

«(3,p)

(

~:;~ ) = «(3, (3)

G(rjJ,rjJ)

=(

~1 FC- 1

1

o -EC- 1

(61)

19

These brackets are identical with the so-called Dirac brackets for this constrained Hamiltonian system. As a consistency check one may use expression (54) and verify (by tedious algebra) that (r,(3)

= -FC- 1 ,

(P,(3)

= EC- 1 .

(62)

Denote the Jacobi field operator (55) by FJ. By varying the matrix elements in it and making use of the variational law 6 lndetG+ = tr(6FJG+), where "tr" includes integration over rp as well as summation over diagonal matrix elements, one easily finds 6lndetG+

= -14>+ C-

1

6C6(rp,rp)drp= -

4>-

L

in which use is made of the formal identity 6( rp, rp )drp final form. We therefore have detG+

+ 6InC(rp), (63)

4>_+

=

II +

const. x

= 1 in passing to the

C-1(rp)

(64)

4>-+

p.[,8,r,p]

=

(detG+)-1/2 = const. x

II +

C 1/ 2(rp)

4>-+

(65) (see eq. (57». The plus sign on the summation symbol in eq. (63) and on the product symbols in eqs. (64) and (65) reflects the fact that C(rp) is unrestricted at rp+ and rp_. 11

Evaluation of the path integral

The path integral (52) now becomes 1+

= N j[dr][dP]

00

II +

j d(3n-l/2r-3/\_1l)(3-3/El)[v'V+X(!3'V"v',4>)][4>.

4>- +

(66)

0

Remarkably, the (3 integration can be carried out exactly. Set (67)

u

z

-llrrp·

(68)

Then each individual (3 integral has a form yielding a Hankel function of order 1/2. Referring to the explicit expression (48) for the function X«(3, r,p, rp) one easily sees that the quantity standing to the right of the product symbol in (66) may be written

20

This lovely result reduces expression (66) simply to

1+

-j

=N

i

e

f+[pr-l£(V'v',4>)lf4>

"'-

(70)

[dr][dp],

which is the path integral for a standard canonical system having 1i as its Hamiltonian. This Hamiltonian yields exactly the same dynamical equations as all the previous action principles do. Its associated Lagrangian is expression (33). Its value is negative in the classically allowed range (see eq. (49)) because the energy due to the expansion (or contraction) of the universe must compensate for the positive energy carried by the contents of the universe, to make the total energy in a compact world vanish. When the quantity under the square root in (49) becomes negative (classically non-allowed region) 1i must be understood as becoming negative imaginary so that the path integral (70) will converge. Note that the measure functional in (70) is just a constant! This is always true for the Hamiltonian path integral of a standard canonical system because the standard Liouville volume element [dr][dp] already gives the correct measure. Note also that given the form (65) for J.t[,B, r,p], which led to this simple result, one could in principle work backwards to infer the measure functional 1'[0, R, 4>, P, II] for the path integral in which the Hamiltonian form of the original action (1), (2) appears. (Here P and II are the momenta conjugate to Rand 4> respectively.) The Hamiltonian 1i (eq. (49)) is a simple enough function of p that the p integrations in (70) can also be performed exactly. Using the identity

f

oo

00

ei(ap+h/p2+c2)dp

I:

= -21 hH~l)1 (c";b 2 2 2 b

-

a 2 ),

(71)

a

one finds, by straightforward algebra,

where

i

e [pr-l£(V,)H4> dp =

12V3n2r2.e-001i~00)1 (.er4»

(72)

.e is the Lagrangian (33). The path integral (70) therefore reduces to

II

1+ = const. x j[dr]

r2.e-001i~00)1 (.cr4».

(73)

4>-+

The Hankel function of order 0, unlike the Hankel function of order 1/2, is not expressible as an exponential, and (73) is not equal to const.

x

j

J.t[r][dr]

II

eiCr 4>

4>-+

=

const.

j

J.t[r][dr]e

if"'+cr4> "'-

(74)

for any ultralocal J.t[r]. This shows the unsuitability of the Lagrangian form of the path integral for our model. The Lagrangian form is probably unsuitable for other mini-superspace models as well and may even be unsuitable for the full field thoery, at least in a compact universe, which means that one should perhaps start with the Palatini form of the action.

21 12

Normalization, Schr6dinger equation and Wheeler-DeWitt equa-

tion The path integral (70) defines a probability amplitude (r + ,1/>+ Ir - ,1/>-)+ for a system with Hamiltonian 1l. But this is a fictitious system, not our original mini-superspace model, for we have got expression (70) by making the false monotonicity assumption. However, before attempting to correct for the assumption, let us look at some of the properties of (r + ,1/>+ Ir - ,1/>- )+. First of all, if this amplitude is properly normalized it will satisfy probability conservation in the guise of the combination law

1 00

(r,l/>lr',I/>')+ =

(r,l/>lr",I/>")+(r",I/>"lr',I/>')+dr", for all 1/>", (75)

together with the boundary condition

(r, I/>Ir', 1/»+

= o(r, r') for all 1/>.

The correct normalization is achieved by choosing mally to be

N in

(76)

expression (70) for(77)

so that each volume element of phase space is divided by h:

drdp

= drdp =

Ii

27rh

drdp when Ii = 1. 27r

(78)

(r, I/>Ir', 1/>')+ also satisfies a Schrodinger equation. By varying the Hamiltonian in the path integral (70) and computing the resulting variation in the 3

integral itself, one can show that the Hamiltonian operator in the Schrodinger equation, which we shall denote by 1lv, is the r-representative of

T1l(r, p, 1/».

(79)

The chronological ordering operation makes expression (79) invariant under point transformations in the (r,p) cotangent bundle (phase space). When the base space of the bundle is I-dimensional, as here, the only invariant way to order the r2p2 term that stands inside the square-root sign of the operator version of (49) is to set it equal to the Laplace-Beltrami operator in a I-dimensional space with metric r- 2 , namely rl/2 prpr l/2 --+ _r 1/ 2 !..-r!..-rl/2 .

8r 8r

(SO)

1l+ is not a polynomial in this operator (as is evident from eq. (49)), but is nonlocal (pseudo-elliptic). One way to try converting the Schrodinger equation (SI) to a local equation is to apply the operator i8/ 81/> again. But this works only if 1lv has no explicit dependence on 1/>. In general one has 2

8 ( r,l/>r,1/> 1")+=t~ .81lv (r,l/>r,1/> 1")++1lv E ( V,'I'V "'I'"' ,'I'"") +, -81/>2 (S2)

22

which is still nonlocal. Only in the case of minimal coupling (V = 0) does (r, cfJlr', cfJ')+ satisfy a local differential equation:

o = (-

:;2

-llv

E)

(r, cfJlr', cfJ')+

(83)

This is, of course, the Wheeler-DeWitt equation for this case, as may be checked by going back and computing the constraint for the original Lagrangian (2). 13

The corrected amplitude

But even in this case one cannot expect the corrected amplitude to satisfy a local differential equation. I shall now decribe what I think is a correct implementation of the instructions implicit in the original path integral (12) (or its Hamiltonian version). I shall make use of the amplitude (r, cfJlr', cfJ')+ as well as the corresponding amplitude (r, cfJlr', cfJ')- obtained by starting with cfJ < cfJ' instead of cfJ > cfJ'. By retracing all the steps of our earlier arguments or simply by noticing that the original Lagrangian (2) is invariant under the replacement ¢ --+ -¢, it is not difficult to see that these amplitudes are transposes of one another:

(r,cfJlr',cfJ')-

= (r',¢>'lr,cfJ)+,

(84)

The Hamiltonian operators appearing in their Schrodinger equations are therefore the negatives of each other: i :cfJ (r, cfJlr', cfJ')± = ±llv{Y', cfJl Y", cfJ')± .

(85)

A corollary of eq. (84) is that the two amplitudes are complex conjugates. They are both, of course, well-defined outside their original ranges (cfJ > cfJ' or cfJ < cfJ'). To keep them in their original ranges we can chop them with step functions. I shall introduce the mixed amplitude

(r, cfJlr', cfJ')/),. = fJ(cfJ - cfJ')(r, cfJlr', cfJ')+ + fJ(cfJ' - cfJ)(r, cfJlr', cfJ')- , (86)

which is symmetric in its arguments. In the minimal-coupling case, and for cfJ =1= cfJ', it satisfies the local Wheeler-DeWitt equation. Now go back to the original path integral, with the action taken in the Hamiltonian form as a functional of a, R, cfJ, P, II, and divide the original time interval [t_ , t+l into smaller intervals [tk , tk-Il where t+ > tl > t2 > ... tN > L with the aim of eventually letting these intervals become arbitrarily small, N going to infinity. Then set r and cfJ equal to fixed values r + ,rl ,r2 , ... , rN , r - , cfJ+ , cfJI , cfJ2 , ... , cfJN , cfJ- at the ends of these intervals. Assume that cfJ(t) is monotonic, either increasing or decreasing, within each interval. Then get rid of the diffeomorphism group by converting to the variables r, p and f3 in each interval. Finally, complete the original functional integration by integrating over rl , r2 , ... , r N , cfJI , cfJ2 , ... , cfJ N·

23

The contribution to the path integral from within each interval is clearly the mixed amplitude (86) for that interval. But the following question arises: What measure should one use in the integration over the r's and ~'s? The original path integral is over R's and ~'s. Therefore one should insert the Jacobian factor b(R)/8(r) (see eq. (31)). This can be computed as follows. From eq. (17) it is easy to see that br(~)/bR(t/)

= b(~(t/) -

~)¢(t/).

(87)

It is straightforward to check that the inverse of this is bR(t)/br(~')

Varying the function

~(t)

= b(~(t) -

~/).

(88)

one obtains the formal expression

b(r) bin b(R)

[ t+ dt j"r"+ b(~(t) - ~)[b/(~(t) - ~)¢(t)b~(t) tP-

t_

+b(~(t) - ~)b¢(t)l

=

[t+ [b/(~(t) _ ~(t))¢(t)b~(t) + b(~(t) - ~(t))b¢(t)ldt.(89) L

The first term inside the final brackets may be regarded as vanishing by symmetry. The second may be simplified by invoking the formal relation b(~(t) - ~(t)) = b(t, t)/¢(t). Setting b(t, t)dt = 1 (cf. eq. (63)), one therefore finds b¢(t)/¢(t)

= bin

II

1¢(t)l,

(90)

whence b(r) b(R)

const. x

II LN N-+oo ZeN) Jo J-oo -00

0

x Ie/> - e/>111/31e/>1 - e/>2 12/ 3Ie/>l - e/>31 1/ 3.. ·Ie/>N-l - e/>'ll/31e/>N - e/>'ll/3 . (93) One expects (or at least hopes!) that the limit N -+ the representation chosen for 1/1~(t)l. 14

00

will be insensitive to

Final remarks

The amplitude (93), which is symmetric in its arguments, does not satisfy any simple combination law. Nor does it satisfy the Wheeler-DeWitt equation other than approximately. The contribution that it receives from that portion of the domain of integration in (93) for which e/>, e/>l , ... ,e/>N and e/>' form a monotonic sequence (cf. eq. (19» is just (r, e/>Ir', e/>') l:>. up to a constant factor. This factor will be unity provided one sets

Simple arguments suffice to give the following very rough estimate of the asymptotic behavior of this function:

ZeN) '" const. x

~(e/21/3)N .

(95)

The contribution to (93) coming from nonmonotonic regions of the domain of integration involves paths that deviate widely from classical, and one expects that destructive interference will keep this contribution small. But even if this is so, and (r, e/>Ir', e/>') is approximately equal to (r, e/>Ir', e/>') l:>., this does not mean that it can be interpreted in any strict sense as a probability amplitude. One can ultimately blame this on the fact that r and r' are not eigenvalues of any diffeomorphism invariant operator, certainly not the operator (17) when fluctuations into nonmonotonic sequences playa role, as they always do. Nevertheless, the statement that the system is at a given point in the (r, e/»-plane is a diffeomorphism invariant statement, and the amplitude (r, tlr', t') must mean something. Here is where the consistent histories 5

view of quantum mechanics can be helpful. Jim Hartle has begun to explore how so-called decoherence functions can be constructed as double functional integrals (or closed-loop functional integrals) of exactly the diffeomorphisminvariant type I have been considering here, but in which the paths would (in the present context) be restricted to pass sequentially through certain fixed regions of the (r, e/» plane. If these regions can be chosen, as part of an exhaustive set, in such a way that the decoherence function has certain well-defined

25 properties mimicking the calculus of joint probabilities, then one can say that "probability" has emerged from the formalism as a phenomenological concept. This is a very exciting program for the future in which, as far as I can see, the Wheeler-DeWitt equation has little or no role to play. 1. A. Peres, Nuovo Cimento 26, 53 (1962).

2. C.J. Isham, Lectures presented at the NATO Advanced Study Institute "Recent Problems in Mathematical Physics," Salamanca, June 15-27, 1992. 3. B.S. DeWitt, Supermanifolds (Second Edition), Cambridge University Press (1992). 4. B.S. DeWitt, in Relativity, Groups and Topology II, eds. B.S. DeWitt and R. Stora, North Holland (1984). 5. J.B. Hartle, "Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime" in Gravitation and Quantizations, eds. B. Julia and J. Zinn-Justin, Les Houches Summer School Proceedings Vol. LVII, North Holland, Amsterdam (1995).

26

PERTURBATIVE DYNAMICS OF QUANTUM GENERAL RELATIVITY JOHN F. DONOGHUE Department of Physics and Astronomy, University of Massachusetts, Amherst, MA 01003 U.S.A. The quantum theory of General Relativity at low energy exists and is of the form of called "effective field theory". In this talk I describe the ideas of effective field theory and the application of this technique to General Relativity.

1

Introduction

The conference organizers originally suggested that the title of this talk be: "Gravity and the Quantum: The view from particle physics". While it is presumptuous for me to claim to speak for all particle physicists, there is in fact a widely held "view" within the particle community that carries an important insight not always appreciated within the gravity community. In this visual analogy, we see clearly a variety of beautiful low-lying hills representing the Standard Model and its applications in known physics. However, there are two sets of clouds on the horizon which ultimately obscure our view. One cloud is located at 1 TeV, just beyond the reach of present accelerators. Beyond this scale, we expect to find the physics which governs electroweak symmetry breaking, with the expectation being that we will uncover new particles and new interactions that change the way that we think of Nature. The other cloud is more ominous, sitting at the Planck scale. Beyond this, we don't know what to expect, but it likely will be something totally new. Neither the Standard Model nor General Relativity is likely to emerge unchanged beyond this scale. So this "view" recognizes that the scenery that we see (our low energy theory) is likely to change if we are ever able to see beyond the "clouds". The insight behind this view suggests a new way of looking at quantum General Relativity. Since we only know that General Relativity is valid at low energy, the key requirement is that the quantum theory can be applied to gravity at present scales. What goes on beyond the Planck scale is a matter of speculation, but gravity and quantum mechanics had better go together at the scales where they are both valid. The good news is that the quantum theory of General Relativity at low energies exists and is well behaved. It is of the form of a type of field theory called Effective Field Theory 1. This is true no matter what the ultimate high-energy theory turns out to be. Given all the work that has gone into quantum gravity, I feel that this is a significant result. The development of effective field theory is an important part of the past decade, and anyone who cares about field theory should learn about it. It has become a standard way of calculating within particle physics, and the way of thinking is widely internalized in the younger generation. In fact, it is would be a reasonably common expectation of young theorists that it is possible that the gravitational effective field theory may turn out to be a better quantum theory than the Standard Model, as the former may extend in validity all the way up to the

27 Planck scale, while the Standard Model will likely be fundamentally modified at 1 TeV. This talk describes some of the features of the effective field theory of General Relativity 2,3. The effective field theory completes a program for quantizing General Relativity that goes back to Feynman and De Witt 4 , and which has received contributions from many researchers over the years. Earlier work focused on quantization and on the divergence structure at high energy. The contribution of effective field theory is to shift the focus back to the low energy where the theory is valid, and to classify the reliable predictions. The low energy quantum effects are distinct from whatever goes on at high energy. Of course, the effective theory does not answer all the interesting questions that we have about the ultimate theory. However, in principle it answers all those questions that we have a right to know with our present state of knowledge about the content of the theory. I will attempt to be clear about the limits of the effective theory as well as its virtues. The outcome of this is that we need to stop spreading the falsehood that General Relativity and Quantum Mechanics are incompatible. They go together quite nicely at ordinary energies. Rather, a more correct statement is that we do not yet know the ultimate high energy theory in Nature. This change in view is important for the gravity community to recognize, because it carries the implication that the ultimate theory is likely to be something new, not just a blind continuation of General Relativity beyond the Planck scale

2

Effective field theory

First let's describe effective field theory in general. Once you understand the basic ideas it is easy to see how it applies to gravity. The phrase "effective" carries the connotation of a low energy approximation of a more complete high energy theory. However, the techniques to be described don't rely at all on the high energy theory. It is perhaps better to focus on a second meaning of "effective", "effective" '" "useful" , which implies that it is the most effective thing to do. This is because the particles and interactions of the effective theory are the useful ones at that energy. An "effective Lagrangian" is a local Lagrangian which describes the low energy interactions. "Effective field theory" is more than just the use of effective Lagangians. It implies a specific full field-theoretic treatment, with loops, renormalization etc. The goal is to extract the full quantum effects of the particles and interactions at low energies. The key to the separation of high energy from low is the uncertainty principle. When one is working with external particles at low energy, the effects of virtual heavy particles or high energy intermediate states involve short distances, and hence can be represented by a series of local Lagrangians. A well known example is the Fermi theory of the weak interactions, which is a local effective Lagrangian describing the effect of the exchange of a heavy W boson. This locality is true even for the high energy portion of loop diagrams. An example of the latter is the high energy portion of the fermion self energy, which is equivalent to a mass counterterm in a local Lagrangian. In contrast, effects that are non-local, where the particles propagate long distances, can only come from the low energy part of the theory.

28 The exchange of a massless photon at low energy can never be represented by a local Lagrangian. From this distinction, we know that we can represent the effects of the high energy theory by the most general local effective Lagrangian. The second key is the energy expansion, which orders the infinite number of terms within this most general Lagrangian in powers of the low energy scale divided by the high energy scale. To any given order in this small parameter, one needs to deal with only a finite number of terms (with coefficients which in general need to be determined from experiment). The lowest order Lagrangian can be used to determine the propagators and low energy vertices, and the rest can be treated as perturbations. When this theory is quantized and used to calculate loops, the usual ultraviolet divergences will share the form of the most general Lagrangian (since they are high energy and hence local) and can be absorbed into the definition of renormalized couplings. There are however leftover effects in the amplitudes from long distance propagation which are distinct from the local Lagrangian and which are the quantum predictions of the low energy theory. This technique can be used in both renormalizable and non-renormalizable theories, as there is no need to restrict the dimensionality of terms in the Lagrangian. (Note that the terminology is bad: we are able to renormalize non-renormalizable theories!) Renormalizable theories are a particularly compact and predictive class of theories. However, many physical effects require non-renormalizable interactions and these need not destroy the quantum theory. In fact, a common calculational device is to isolate the relevant interactions only, even if this implies a non-renormalizable theory, and to use the techniques of effective field theory to perform a simpler calculation than if one were to compute using the full theory. This is done in Heavy Quark Effective Theory 5 as well as in the theory of electroweak radiative corrections. The effective field theory which is most similar to general relativity is chiral perturbation theory 6, which describes the theory of pions and photons which is the low energy limit of QeD. The theory is highly nonlinear, with a lowest order Lagrangian which can be written with the exponential of the pion fields 2

2

F m c = F2i Tr (V pUVPU t ) + ~Tr (U + U t )

u

=

exp

(i ri;;x))

(1)

with ri being the SU(2) Pauli matrices and F" = 92.3MeV being a dimensionful coupling constant. This shares with general relativity the dimensionful coupling, the non-renormalizable nature and the intrinsically nonlinear Lagrangian. This theory has been extensively studied theoretically, to one and two loops, and experimentally. There are processes which clearly reveal the presence of loop diagrams. In my talk, I displayed some of the predictions and experimental tests of chiral perturbation theory, most of which were taken from the book published with my co-authors 1. The point was to illustrate the fact that effective field theory is not just an idea, but is a practical tool that is applied in real-world physics. In a way, chiral perturbation theory is the model for a complete non-renormalizable effective field theory in the same way that QED serves as a model for renormalizable field theories.

29

3

Overview of the gravitational effective field theory

At low energies, general relativity automatically behaves in the way that we treat effective field theories. This is not a philosophical statement implying that there must be a deeper high energy theory of which general relativity is the low energy approximation. Rather, it is a practical statement. Whether or not general relativityis truly fundamental, the low energy quantum interactions must behave in a particular way because of the nature of the gravitational couplings, and this way is that of effective field theory. The Einstein action, the scalar curvature, involves two derivatives on the metric field. Higher powers of the curvature, allowed by general covariance, involve more derivatives and hence the energy expansion has the form of a derivative expansion. The higher powers of the curvature in the most general Lagrangian do not cause problems when treated as low energy perturbations 7. The Einstein action is in fact readily quantized, using gauge-fixing and ghost fields ala Feynman, DeWitt, Faddeev, Popov 4, The background field method used by 'tHooft and Veltman 8 is most beautiful in this context because it allows one to retain the symmetries of general relativity in the background field, while still gauge-fixing the quantum fluctuations. The applications of these methods allow the quantization of general relativity in as straight-forward a way as QeD is quantized. The problem with the field theory program comes not at the level of quantization, but in attempting to make meaningful calculations. The dimensionful nature of the gravitational coupling implies that loop diagrams (both the finite and infinite parts) will generate effects at higher orders in the energy expansion 9. In previous times when we only understood renormalizable field theory, this was a problem because the divergences could not be dealt with by a renormalization of the original Lagrangian. However, in effective field theory, one allows a more general Lagrangian. Since the divergences come from the high energy portion of loop integrals, they will be equivalent to a local term in a Lagrangian. Since the effective Lagrangian allows all terms consistent with the theory, and each term is governed by one or more parameters describing its strength, there is a parameter available corresponding to each divergence. We absorb the high energy effects of the loop diagram into a renormalized parameter, which also contains other unknown effects from the ultimate high energy theory. The one and two loop counterterms for graviton loops are known 8,10 and, as expected, go into the renormalization of the coefficients in the Lagrangian. However, these are not really predictions of the effective theory. The real action comes at low energy. How in practice does one separate high energy from low? Fortunately, the calculation takes care of this automatically, although it is important to know what is happening. Again, the main point is that the high energy effects share the structure of the local Lagrangian, while low energy effects are different. When one completes a calculation, high energy effects will appear in the answer in the same way that the coefficients from the local Lagrangian will. One cannot distinguish these effects from the unknown coefficients. However, low energy effects are anything that has a different structure. Most often the distinction is that of analytic versus non-analytic in momentum space. Analytic expressions can be Taylor expanded in

30

the momentum and therefore have the behavior of an energy expansion, much like the effects of a local Lagrangian ordered in a derivative expansion. However, nonanalytic terms can never be confused with the local Lagrangian, and are intrinsically non-local. Typical non-analytic forms are J _q2 and ln( _q2). These are always consequences of low energy propagation. Having provided this brief overview of the way that effective field theory may be applied to general relativity, let me be a bit more explicit about some of these steps. 4

Then energy expansion in general relativity

What is the rationale for choosing the gravitational action proportional to R and only R? It is not due to any symmetry and, unlike other theories, cannot be argued on the basis of renormalizability. However physically the curvature is small so that in most applications R2 terms would be yet smaller. This leads to the use of the energy expansion in the gravitational effective field theory. There are in fact infinitely many terms allowed by general coordinate invariance, i.e.,

s=

!

d"x.,f9 { A +

:2

R + c1 R

2

+ C2 R,.",RI-'V + ... + Cmatter}

(2)

Here the gravitational Lagrangians have been ordered in a derivative expansion with A being of order 8 0 , R of order 8 2 , R2 and Rl-'vRI-'V of order 8 4 etc. Note that in four dimensions we do not need to include a term Rl-'va{3Rl-'va{3 as the Gauss Bonnet theorem allows this contribution to the action to be written in terms of R2 and Rl-'vRl-'v. The first term in Eq.21 , i.e., A, is related to the cosmological constant, A = -87rGA. This is a term which in principle should be included, but cosmology bounds 1A 1< 1O-56 cm -2, 1A 1< 1O-46GeV 4 so that this constant is unimportant at ordinary energies. We then set A = 0 from now on. In contrast, the R2 terms are able to be shown to be unimportant in a natural way. Let us drop Lorentz indices in order to focus on the important elements, which are the numbers of derivatives. A R + R2 Lagrangian 2

2 C= -R+cR K,2

(3)

has an equation of motion which is of the form

(4) The Greens function for this wave equation has the form

G(x) (5)

31

The second term appears like a massive scalar, but with the wrong overall sign, and leads to a short-ranged Yukawa potential (6)

The exact form has been worked out by Stelle 11 , who gives the experimental bounds 74 Cl, C2 < 10 . Hence, if Ci were a reasonable number there would be no effect on any observable physics. [Note that if C '" 1, J",2 C '" 1O- 35 mj. Basically the curvature is so small that R2 terms are irrelevant at ordinary scales. As a slightly technical aside, in an effective field theory we should not treat the R2 terms to all orders, as is done above in the exponential of the Yukawa solution, but only include the first corrections in ",2C. This is because at higher orders in ",2c we would also be sensitive to yet higher terms in the effective Lagrangian (R 3 , R4 etc.) so that we really do not know the full r ~ 0 behavior. Rather, for J",2 C small we can note the Yukawa potential becomes a representation of a delta function

e- r / VJC ~ 41l"",2ct5 3 (f') r The low energy potential then has the form ---

(7)

(8)

R2 terms in the Lagrangian lead to a very weak and short range modification to the gravitational interaction. Thus when treated as a classical effective field theory, we can start with the more general Lagrangian, and find that only the effect of the Einstein action, R, is visible in any test of general relativity. We need not make any unnatural restrictions on the Lagrangian to exclude R2 and R/wRI-'V terms. 5

Quantization

There is a beautiful and simple formalism for the quantization of gravity. The most attractive variant combines the covariant quantization pioneered by Feynman and De Witt 4 with the background field method introduced in this context by 't Hooft and Veltman 8. The quantization of a gauge theory always involves fixing a gauge. This can in principle cause trouble if this procedure then induces divergences which can not be absorbed in the coefficients ofthe most general Lagrangian which displays the gauge symmetry. The background field method solves this problem because the calculation retains the symmetry under transformations of the background field and therefor the loop expansion will be gauge invariant, retaining the symmetries of general relativity. Consider the expansion of the metric about a smooth background field 9I-'v(X), (9)

32

Indices are now raised and lowered with g. The Lagrangian may be expanded in the quantum field hj.£11 8. 2 = -.;gR K,2

=

C(1) 9

.J9 {2.R + C(1) K,2 9

+ C(2) + ... } 9

hj.£11 [gj.£11 R _ 2Rj.£II] K,

C(2)

=

9

21 D Ot h j.£11 D Ot hj.£1I -

1 h D Oth + DOth D {3h Ot/3 2DOt

+R

-DOt h j.£{3D{3hj.£Ot

(10)

(!h2 2 -!h 2 j.£11 hj.£lI)

+Rj.£11 (2h Aj.£h IlOt - hhj.£lI) Here DOt is a covariant derivative with respect to the background field. The total set of terms linear in hj.£11 (including those from the matter Lagrangian) will vanish if gj.£11 satisfies Einstein's equation. We are then left with a quadratic Lagrangian plus interaction terms of higher order. However, the quadratic Lagrangian cannot be quantized without gauge fixing and the associated Feynman-DeWitt-Fadeev-Popov ghost fields. In this case, we would like to impose the harmonic gauge constraint in the background field, and can choose the constraint 8

(11) where 'TJOt{3tj.£ Ot t ll {3

= gj.£11

(12)

This leads to the gauge fixing Lagrangian 8 ll

Cgf = .J9 { (D hj.£11 -

~Dj.£h\)

(DtThj.£tT -

~Dj.£htTtT) }

(13)

Because the gauge constraint contains a free Lorentz index, the ghost field will carry a Lorentz label, i.e., they will be fermionic vector fields. After a bit of work the ghost Lagrangian is found to be Cgh

= .J9'TJ*j.£ [DADAgj.£1I -

Rj.£II] 'TJII

(14)

The full quantum action is then of the form

s=

f at sV9 { :2

R-

~hOt{3DOt/3,-Y5 h-Y5

+ 'TJ*j.£ {DADAgj.£1I - Rj.£II} 'TJII + O(h 3 )}

(15)

33

6

Renormalization

The one loop divergences of gravity have been studied in two slightly different methods. One involves direct calculation of the Feynman diagrams with a particular choice of gauge and definition of the quantum gravitational field 12. The background field method, with a slightly different gauge constraint, allows one to calculate in a single step the divergences in graphs with arbitrary numbers of external lines and also produces a result which is explicitly generally covariant 8. In the latter technique one expands about a background spacetime glJ. v , fixes the gauge as we described above and collects all the terms quadratic in the quantum field hlJ.v and the ghost fields. For the graviton field we have

!

Z[g]

= =

[dhlJ.v]exP{i! crX.JY{:2R+hlJ.vDIJ.Va,Bha,B}

detDlJ.va,B expTrln(DlJ.va,B)

(16)

where DlJ.va,B is a differential operator made up of derivatives as well as factors of the background curvature. The short distance divergences of this object can be calculated by standard techniques once a regularization scheme is chosen. Dimensional regularization is the preferred scheme because it does not interfere with the invariances of general relativity. First calculated in this scheme by 't Hooft and Veltman 8, the divergent term at one-loop due to graviton and ghost loops is described by a Lagrangian (div) _

Clloop

-

1 { 1 -2 7 - - v} 811" 210 120 R + 20 RlJ.vRIJ.

(17)

with 10 = 4 - d. Matter fields of different spins will also provide additional contributions with different linear combinations of R2 and RlJ.v RlJ.v at one loop. The fact that the divergences is not proportional to the original Einstein action is an indication that the theory is of the non-renormalizable type. Despite the name, however, it is easy to renormalize the theory at any given order. At one loop we identify renormalized parameters 1

(r)

c1 (r)

c2

=

Cl + 96011"210 7 C2 + 16011"2 10

(18)

which will absorb the divergence due to graviton loops. Alternate but equivalent expressions would be used in the presence of matter loops. A few comments on this result are useful. One often hears that pure gravity is one loop finite. This is because the lowest order equation of motion for pure gravity is RlJ.v = 0 so that the O(R2) terms in the Lagrangian vanish for all solutions to the Einstein equation. However in the presence of matter (even classical matter)

34

this is no longer true and the graviton loops yield divergent effects which must be renormalized as described above. At two loops, there is a divergence in pure gravity which remains even after the equations of motion have been used 10. ddiv) 2loop

=

209/\:2 ~Ra{3 R'Yo R'T/u 2880(1611'2) € 'YO 'T/U a{3

(19)

For our purposes, this latter result also serves to illustrate the nature of the loop expansion. Higher order loops invariably involve more powers of /\: which by dimensional analysis implies more powers of the curvature or of derivatives in the corresponding Lagrangian (Le., one loop implies R2 terms, 2 loops imply R3 etc.). The two loop divergence would be renormalized by absorbing the effect into a renormalized value of a coupling constant in the O(R3) Lagrangian. 7

Quantum predictions in an effective theory

At this stage it is important to be clear about the nature of the quantum predictions in an effective theory. The divergences described in the last section come out of loop diagrams, but they are not predictions of the effective theory. They are due to the high energy portions of the loop integration, and we do not even pretend that this portion is reliable. We expect the real divergences (if any) to be different. However the divergences do not in any case enter into any physical consequences, as they absorbed into the renormalized parameters. The couplings which appear in the effective Lagrangian are also not predictions of the effective theory. They parameterize our ignorance and must emerge from an ultimate high energy theory or be measured experimentally. However there are quantum effects which are due to low energy portion of the theory, and which the effective theory can predict. These come because the effective theory is using the correct degrees of freedom and the right vertices at low energy. It is these low energy effects which are the quantum predictions of the effective field theory. It may at first seem difficult to identify which components of a calculation correspond to low energy, but in practice it is straightforward. The effective field theory calculational technique automatically separates the low energy observables. The local effective Lagrangian will generate contributions to some set of processes, which will be parameterized by a set of coefficients. If, in the calculation of the loop corrections, one encounters contributions which have the same form as those from the local Lagrangian, these cannot be distinguished from high energy effects. In the comparison of different reactions, such effects play no role, since we do not know ahead of time the value of the coefficients in.c. We must measure these constants or form linear combinations of observables which are independent of them. Only loop contributions which have a different structure from the local Lagrangian can make a difference in the predictions of reactions. Since the effective Lagrangian accounts for the most general high energy effects, anything with a different structure must come from low energy. A particular class of low energy corrections stand out as the most important. These are the nonlocal effects. In momentum space the nonlocality is manifest by a

35

nonanalytic behavior. Nonanalytic terms are clearly distinct from the effects of the local Lagrangian, which always give results which involves powers of the momentum.

8

Examples

A conceptually simple (although calculationally difficult) example is graviton-graviton scattering. This has been calculated to one-loop in an impressive paper by Dunbar and Norridge 13 using string based methods. Because the reaction involves only the pure gravity sector, and R/-IV = 0 is the lowest order equation of motion, the result is independent of any of the four-derivative terms that can occur in the Lagrangian (R 2 or RlJ.vRIJ.V). Thus the result is independent of any unknown coefficient to one loop order. Their result for the scattering of positive helicity gravitons is

A(++ --t ++)

84

=

87rG stu { 1

+

~[

(tln( ~u) In( 7) + uln(~t) In(7) + s In(~t) In( ~u))

+ In(!) tu(t -6 u) (341(t 4 + u 4) + 1609(eu + u 3t) + 2566t2U2) + +

U 60s (In2(!) + 7r 2) tu(t + 2u)(u + 2t) (2t4 + 2u4 + 2t3u + 2u3t _ t2U2) U 2s7 3:~s5 (1922(t 4 + u 4) + 9143(t3u + u 3t) + 14622t2u 2) l} (20)

where s = (PI + P2)2, t = (PI - P3)2, U = (PI - P4)2, (8 + t + u = 0) and where I have used 8 as an infrared cutoff 14. One sees the non-analytic terms in the logarithms. Also one sees the nature of the energy expansion in the graviton sector - it is an expansion in G E2 where E is a typical energy in the problem. I consider this result to be very beautiful. It is a low energy theorem of quantum gravity. The graviton scattering amplitude must behave in this specific fashion no matter what the ultimate high energy theory is and no matter what the massive particles of the theory are. This is a rigorous prediction of quantum gravity. The other complete example of this style of calculation is the long distance quantum correction to the gravitational interaction of two masses 3,15. This is accomplished by calculating the vertex and vacuum polarization corrections to the interaction of two heavy masses. In addition to the classical corrections 16, one obtains the true quantum correction

v;Ipr () r

= _ Gm l m2 r

[1 _13530+ 2N

v

7r 2

Gh

r 2 c3

+ ...

]

(21)

for a specific definition of the potential. Note that the result is finite and independent of any parameters. This is easy to understand once one appreciates the structure of effective field theory. The divergences that occur in the loop diagrams all go into the renormalization of the coefficients in the local Lagrangian, as we displayed above. Since these terms in the Lagrangian yield only delta-function modifications to the potential, they cannot modify any power-law correction that

36

survives to large distance. Only the propagation of massless fields can generate the nonanalytic behavior that yields power-law corrections in coordinate space. Since the low energy couplings of massless particles are determined by Einstein's theory, these effects are rigorously calculable. Note that this calculation is the first to provide a quantitative answer to the question as to whether the effective gravitational coupling increases or decreases at short distance due to quantum effects. While there is some arbitrariness in what one defines to be Gell, it must be a universal property (this eliminates from consideration the Post-Newtonian classical correction which depends on the external masses) and must represent a general property of the theory. The diagrams involved in the above potential are the same ones that go into the definition of the running coupling in QED and QeD and the quantum corrections are independent of the external masses. If one uses this gravitational interaction to define a running coupling one finds G e (r) = G [1 _ 135 + 2Nv Gh] (22) II 30rr2 r 2 c3 The quantum corrections decrease the strength of gravity at short distance, in agreement with handwaving expectations. (In pure gravity without photons or massless neutrinos, the factor 135 + 2Nv is replaced by 127.) An alternate definition including the diagrams calculated in 15 has a slightly different number, but the same qualitative conclusion. The power-law running, instead of the usual logarithm, is a consequence of the dimensionful gravitational coupling. These two results do not exhaust the predictions of the effective field theory of gravity. In principle, any low energy gravitational process can be calculated 17. The two examples above have been particularly nice in that they did not depend on any unknown coefficients from the general Lagrangian. However it is not a failure of the approach if one of these coefficients appears in a particular set of amplitudes. One simply treats it as a coupling constant, measuring it in one process (in principle) and using the result in the remaining amplitudes. The leftover structures aside from this coefficient are the low energy quantum predictions. 9

Limitations and the high energy regime

The effective field theory techniques can be applied at low energies and small curvatures. The techniques fail when the energy/curvature reaches the Planck scale. There is no known method to extend such a theory to higher energies. Indeed, even if such a technique were found, the result would likely be wrong. In all known effective theories, new degrees of freedom and new interactions come into play at high energies, and to simply try to extend the low energy theory to all scales is the wrong thing to do 18. One needs a new enlarged theory at high energy. However, many attempts to quantize general relativity ignore this distinction and appear misguided from our experience with other effective field theories. While admittedly we cannot be completely sure of the high energy fate of gravity, the structure of the theory itself hints very strongly that new interactions are needed for a healthy high energy theory. It is likely that, if one is concerned with only pure general relativity, the effective field theory is the full quantum content of the theory.

37

10

Summary

The quantum theory of general relativity at low energy has turned out to be of the form that we call effective field theories. The result is a beautiful theory that incorporates general coordinate invariance in a simple way, and which has a known methodology for extracting predictions. The theory fits well with the other ingredients of the Standard Model. It is common, but wrong, to imply that general relativity differs for the other interactions because it has no known quantum theory. As we have seen, the quantum theory exists at those scales where General Relativity is reliably thought to apply. Many of the most interesting questions that we ask of quantum gravity cannot be answered by the effective field theory. This is a warning that these questions require knowledge of physics beyond the Planck scale. Since physics is an experimental science, thoughts about what goes on at such a high scale may remain merely speculation for many years. However, it is at least reassuring that the ideas of quantum field theory can successfully be applied to General Relativity at the energy scales that we know about. References 1. Introductions to the ideas of effective field theory can be found in:

S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995). J.F. Donoghue, E. Golowich and B.R. Holstein, Dynamics of the Standard Model (Cambridge Univ. Press, Cambridge, 1992). H. Georgi, Weak Interactions and Modern Particle Theory (Benjamin/Cummings, Menlo Park, 1984). S. Weinberg, Physica (Amsterdam) 96A, 327 (1979). J. Gasser and H. Leutwyler, Nucl. Phys. B2S0, 465 (1985). J.F. Donoghue, in Effective Field Theories of the Standard Model, ed. by U.G. Meissner (World Scientific, Singapore, 1992) p. 3. J. Polchinski, Proceedings of the 1992 TASI Summer School, ed. by J. Harvey and J. Polchinski (World Scientific, Singapore, 1993) p.235 . A. Cohen, Proceedings of the 1993 TASI Summer School, ed. by S. Raby (World Scientific, Singapore, 1994), p.53 . Plenary talks by H. Leutwyler and S. Weinberg in Proceedings of the XXVI International Conference on High Eneryy Physics, Dallas 1992, ed. by J. Sanford (AlP, NY, 1993) pp. 185, 346. D.B. Kaplan, Effective field theories, lectures at the 7th Summer School in Nuclear Physics Symmetries, Seattle 1995, nucl-th/9506035. A.V. Manohar, Effective field theories, Schladming lectures 1996, hepph/9606222. 2. J.F. Donoghue, Phys. Rev. DSO, 3874 (1994). J.F. Donoghue, Introduction to the effective field theory description of gravity, in Advanced School on Effective Theories, ed. by F. Cornet and M. J. Herrero (World Scientific, Singapore, 1996) gr-qc/9512024.

38

3. J.F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994). 4. R.P. Feynman, Acta. Phys. Pol. 24, 697 (1963); Caltech lectures 1962-63. B.S. De Witt, Phys. Rev.160, 1113 (1967), ibid. 162, 1195, 1239 (1967). L. D. Faddeev and V.N. Popov, Phys. Lett. 25B,29 (967). 5. H. Georgi, Phys. Lett. B240, 477 (1990). T. Mannel, Effective Theory for Heavy Quarks, Schladming lectures, 1996, hep-ph/9606299. 6. Most of the references in Ref.[I] discuss chiral perturbation theory. Other sources include: J. F. Donoghue, Chiral symmetry as an experimental science,. in Medium Energy Antiprotons and the Quark-Gluon Structure of Hadrons, ed. by R. Landua, J.-M. Richard and L. Klapish (Plenum, N.Y., 1991) p. 39. J. Bijnens, G. Ecker and J. Gasser, hep-ph/9411232. A. Pich, Rept. Prog. Phys. 58,563 (1995). G. Ecker, Erice lectures, hep-ph/9511412 . 7. J. Simon, Phys. Rev. D41, 3720 (1990); 43, 3308 (1991). 8. 't Hooft and M. Veltman, Ann. Inst. H. Poincare A20, 69 (1974). M. Veltman, in Methods in Field Theory Proc. of the Les Houches Summer School, 1975, ed. by R. Balian and J.Zinn-Justin (North Holland, Amsterdam, 1976). 9. J.F. Donoghue and T. Torma, Phys. Rev. D54, 4963 (1996). 10. M. Goroff and A. Sagnotti, Nucl. Phys. B266, 799 (1986). 11. K.S. Stelle, Gen. ReI. Grav. 9, 353 (1978). 12. D.M. Capper, G. Leibrandt and M. Ramon Medrano, Phys. Rev. D8, 4320 (1973). M.R. Brown, Nucl. Phys. B56, 194 (1973). D.M. Capper, M.J. Duff and L. Halpern, Phys. Rev. DIO, 461 (1944). S.Deser and P. van Niewenhuizen, Phys. Rev. Lett. 32, 245 (1974); Phys. Rev. DIO, 401 411 (1974). S.Deser, H.-S. Tsao and P. van Niewenhuizen, Phys. Rev. DIO, 3337 (1974). 13. D. Dunbar and P. Norridge, NucI. Phys. B433, 181 (1995). 14. J. F. Donoghue and T. Torma, in preparation. 15. H. Hamber and S. Liu, Phys. Lett. B357, 51 (1995), I. Muzinich and S. Vokos, Phys. Rev. D52, 3472 (1995), A. Akhundov, S. Bellucci and A. Shiekh, Phys. Lett. B395, 16 (1997). 16. Y. Iwasaki, Prog. Theo. Phys. 46, 1587 (1971). D. Boulware and S. Deser, Ann. Phys. 89, 193 (1975). S. Gupta and S. Radford, Phys. Rev. D21, 2213 (1980). 17. Hawking radiation can also be calculated in a way that appears consistent with the effective theory and independent of the high energy cut-offs. H. Hambli and C. Burgess, Phys. Rev. D53, 5717 (1996). J. Polchinski, hep-th/9507094 . K. Fedenhagen and R. Haag, Comm. Math. Phys. 127, 273 (1990). T. Jacobsen, this conference. 18. In the question and answer period, Bryce De Witt asked a question that at first appears only marginally related, but in fact goes to the heart of the

39

reasoning of effective field theory. He imagined that it has been proven that pure QED is a perfectly complete and consistent theory in every way satisfying to mathematical physics, and asked if we should care about such a result. Effective field theory would say that as physicists we would not care much about this demonstration since we know that QED does not exist in isolation in the world but is part of a different, larger theory. Even that larger theory is likely not the full story. That these theories work at present scales is the fascinating and important result. This illustrates a difference between physics and mathematics.

40

BROKEN SYMMETRY: APPLYING THE METHOD OF THE SUPERCONNECTION FOR RIEMANNIAN GRAVITY YUVAL NE'EMAN B Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, Israel 69978

Center for Particle Physics, University of Texas, Austin, Texas 78712

Abstract A superconnection is a supermatrix whose even part coincides with the gauge-potential one-forms of a local gauge group, while the odd parts represent the (O-form) Higgs fields; the overall grading is thus odd in both cases. The simple rank 3 supergroup P( 4, R) in Kac' classification of the simple supergroups, with even subgroup 8L(4, R), spontaneously breaks 8L(4, R) as gauge group, leaving just local 80(1,3) unbroken. As a result, post-Riemannian 8KY gravity yields Einstein's theory as a low-energy (longer range) effective theory. The theory is renormalizable and may be unitary. The model can be derived in noncommutative geometry, applying the Mainz-Marseilles modification of the Connes construction. Superconnections and the Electroweak BU(2/1)

1

My invited contribution to this Symposium was meant to deal with Symmetry as an approach to Quantum Gravity. Rather than reviewing past results, I thought it would be more interesting to discuss one new approach, Noncommutative Geometry (NCG), which might open new vistas. In fact, I shall present a description of Einsteinian Gravity as a spontaneously broken SKY SL( 4,R) model, applying Quillen's Superconnection approach, though I shall also discuss its derivation in Cannes' approach (based on the matter fields), except that the NCG form calculus we use is the modified one, introduced by the Mainz-Marseilles collaboration, in their rederiving of the SU(2/1) Electroweak superconnection The 8uperconnection was introduced by Quillenl in the mathematical literature. It is a supermatrix, belonging to a given supergroup B, valued over elements belonging to a Grassmann algebra of forms. In its physics role, the even part of the supermatrix is valued over the gauge-potentials of the even subgroup G c B, (one-forms B~dxJ.t on the base manifold of the bundle, realizing the "gauging" of G). The odd part of the supermatrix, representing the quotient B/G = H c B, is valued over zero-forms in that Grassmann algebra, physically the Higgs multiplet q,H(X), in a spontaneously broken G gauge theory;4>(x) E q,H(X), < OI4>(x)IO ># 0, thus leaving only a subgroup F c G, [F, 4>1 = 0 as the low-energy residual local symmetry. The first physical example of a superconnection preceeded Quillen's theory. This was our BU(2/1) (supergroup) proposal for an algebraically irreducible electroweak unification 3 ,4. Lacking Quillen's generalized formulation, the model appeared to suffer from spin-statistics interpretative complications for the physical B

Wolfson Distinguished Chair in Theoretical Physics

41

fields. The structural Z2 grading of Lie superalgebras, as previously used in Physics (i.e. in supersymmetry) corresponded to Bose/Fermi transitions, so that invariance under the supergroup represented symmetry between bosons and fermions. Here, however, though the superconnection itself does fit the quantum statistics ansatz, this is realized through the order of the forms in the Grassmann algebra, rather than through the quantum statistics of the particle Hilbert space! (both the W:, AI' on the one hand, and the Higgs field iP H on the other, are bosons). Moreover, the matter fields' (leptons or quarks) Hilbert space is Z2-graded by fermion chiralities. The internal quantum numbers set by the SU(2/1) assignments do display a perfect fit with the phenomenology (including the provision that integer charges - leptons come in three states, e.g. [vL,eL/eRj, whereas fractional charges - quarks - come in four, e.g. [UR/UL, dL/d R]) , but they obviously do not lend themselves to a quantum statistics grading ansatz. It was noted 5 that the method appears to apply to a large set of spontaneously broken symmetries, global or local. The example of the Goldstone-Nambu breaking of (global) chiral SU(3) I8l SU(3) was treated in detail in ref. 5, the relevant supergroup being the Q(3) of Kac' classification6 of the simple superalgebras. This is an exceptional supergroup we had encountered earlier7 , precisely because of its physical relevance. The second development was an improved understanding of both the physics and the mathematics of the juxtaposition of the two graded systems - on the one hand, the supergroup as represented by its supermatrices and on the other hand the Grassmann algebra over which it is valued 8 . Tied to the traditional YangMills derivation, however, our Grassmann-even elements, in the group-odd sector of the superconnection, started with two-forms, thus missing the desirable zero-forms, which Quillen could freely postulate. In order to include a fitting scalar field, our 1982 solution used the extended forms system, taken over the entire bundle, in the geometrical interpretation of BRS - i.e. the Higgs was a physical partner to ghosts in a scalar supermultiplet. Our final treatment, with S. Sternberg9 , availed itself of the advantage deriving from Quillen's generalized formalism, also applying the method to a further unification lO , including QCD and a (2k) generations structure, using SU(5 + k/1). An important 'technical' point related to the multiplication of supermatrices (we refer the reader to ref. 10 for precise algorithms). The next installment came from Connes' noncommutative geometry (NCG), generalizing to discrete geometries some geometrical concepts (such as distances) till then defined only for continuous spaces. Connes and Lott l1 used the new formalism to reproduce the electroweak theory, providing it with a geometric derivation: the base manifold is Z2 I8l M 3 / l = ML EEl M R, where Z2 is a discrete space containing just two points L, R representing chiralities and M3/l is Minkowski spacetimE!'. NCG defines a space by the functions and Hilbert space states living on it and the operators acting on that Hilbert space. Here, parallel-transport within ML (or within M R ) is performed by D = d + B(G), with B standing for the relevant gauge potentials and G = SU(2)w x U(l)y. Moving, however, from a state sitting over a point in ML (say vL(x)) to one sitting over a point in MR (say eR(x)) requires a scalar "connection" iP H (x). In this case, its G quantum numbers are entirely fixed by the matter fields' selected assignments; this includes the Lorentz scalar

Z2,

bThe authors of ref. 11 work in Euclidean M4

42

nature of iP H, due to its having to relate e.g. e L to e R in a Yukawa term (the 'Yo is provided for by the Dirac operator, acting on the Hilbert space, in the definition of the noncommutative geometry). The link with our 8U(2/1) superconnection was provided by Coquereaux, Scheck and coworkers of the "Marseilles-Mainz group,,12-16. They found that by slightly modifying the Connes axioms, su(2/1) emerges naturally as the superalgebra of the form-calculus over the discrete Z2 of the chiralities, while the superconnection for the product space Z2®M3/1 is an SU(2/1) group-element. This also finally exorcizes the apparent difficulty with the non-spin-statistics grading of the matter fields and explains how the grading can be related to chiralities instead. Moreover, the paralleltransport operator is found to require an additional "matrix derivative" 8H, relating "twin" states in ML and M R , such as eL and eR, etc .. (this is the role of (3 in the Dirac 'YIJ calculus). With this additional term, the curvature-squared Lagrangian R 1\ .. R for 8 = 8U(2/1) contains the complete Weinberg-Salam Lagrangian. Indeed, R = Ra + (1/2){iPH' iP H } + DaiPH + 8HiPH, with R(G) = dB + (1/2)B 1\ B. In squaring, the second term in R provides for the AiP4 and the fourth provides for the negative mass-squared piece of the Higgs potential.

2

Riemannian Gravity Derived from a Broken SKY SL( 4,R) Gravity

The interest in deriving Einstein's Riemannian theory through the spontaneous symmetry breakdown of a non-Riemannian theory stems from quantum considerations. First, the quantization of gravity implies spacetime quantization at Planck energies (where the Compton wavelength is also the Schwarzschild radius, (h/27rmc) = 2Gm/c2). This quantization, in itself, represents a departure from Riemannian geometry. Secondly, the addition in the Lagrangian of terms quadratic in the curvatures renders the theory finite (the new terms dominate at high-energy and are dimensionless in the action); however, it is nonunitary, due to the appearance of p-4 propagators. These are present because of the Riemannian condition DglJv = 0, relating the connection r to the metric glJv (the Christoffel formula). Thus r ::::= 8g and R = dr + (1/2)rr ::::= (8)2g + (8g)2 and R2 will involve p4 terms in momentum space and thus p-4 propagators. These can then be rewritten as differences between two S-matrix poles - one of which is then a ghost, due to the wrong sign of its residue. It seems therefore worth trying to reconstruct gravity so that the Riemannian condition will only constrain the low-energy end of the theory, as an effective result in that regime. The high-energy theory, i.e. prior to symmetry breakdown, should have as its anholonomic (gauge) group the metalinear 8 L( 4, R). We have investigated a modeP7-20 based on either 8L(4, R) or GL(4, R), containing the Stephenson-Kilmister-Yang (SKY) Lagrangian 21 - 23 plus a term linear in the curvature, and proved the Yang-Mills-like renormalizability and BRST invariance of the quantum Lagrangian. Whether the theory is unitary is not known at this stage, due to the presence of a p-4 term as input in the gauge-fixing term of the quantum Lagrangian 19 - 2o . In such theories, (a) the G = 8 L( 4, R)-invariant R( G) 1\ .. R( G) SKY Lagrangian has to have its symmetry broken by a Higgs field corresponding to an 8 L( 4, R) multiplet containing a Lorentz-scalar component, to ensure that F = 80(1,3). In

43

the algebraic structure we use (the superalgebra of S = P(4, R)), this includes a metric-like symmetric tensor (a, b = 0, .. 3 are anholonomic indices supporting the local action of S and its subgroups) CI> {ab} (x), and the Lorentz scalar is given by c/J = CI>abTJab, where "lab is either the trace (for Euclidean signature situations) or the Minkowski metric. Thus < 01c/J(x)IO >:/= 0. (b) Those components of the connection r~(x) which serve to gauge GIF = SL(4,R)ISO(I,3) should acquire masses in the spontaneous breakdown procedure. As in the electroweak case, we should have in the Higgs multiplet, components which - in the Unitary gauge - will have become the longitudinal (spin) components of the (now massive) G I F elements of the connection. In our construction, these are precisely the 9 components of CI> {ab}, after removal of the trace (or Minkowski-trace). (c) Any remaining components of CI>H(X) should acquire masses and exist as free particles. In the P(4,R) model, CI>H(X) = CI>{ab}(X) EB CI>[abj(X), i.e. there is, in addition, an antisymmetric field CI>[abj(X), which indeed acquires a Planck-scale mass. 3

The Simple Lie Superalgebra p(4, R) and its Superbrackets

The defining representation of the generating superalgebra of the P( 4, R) supergroup is an 8 x 8 matrix, divided into quadrants. I and IV carry the sl( 4, R) algebra, with I in the covariant representation ~f (a, b = 0,1,2,3) and the tildes indicate tracelessnes tr:Et = 0) and IV in the contravariant, i.e. IV = _IT (T indicates transposition). In the off-diagonal quadrants,

II

= ~i} carries the 10

symmetric matrices of gl( 4, R) and II I = ~~l carries its 6 antisymmetric matrices. are even, representing the There are thus altogether 31 generators, of which 15 action of sl(4, R) and 16 Nt are odd, of which 10 are the symmetric N+ = T, and 6 the antisymmetric N- = M,exhausting the set of generators of gl(4, R) (we use the notation of ref. 24, i.e. the T, M are the shears and Lorentz generators, respectively) . The simple superalgebra is thus given as,

Qf

(

I~I

I II II) IV III IV

and Qab

= = = =

~ii_

Tab

~[a

bj

-(~iiii)T

-

(1)

Mab ..1.Q ii ii

- T

(~aiih EB -(~~hv

N"J, .- (~i})II N;;b

tQ iib

b b}

~{a

(2)

(~~l)JII

To formulate the super-Lie bracket, we chose to replace the two-index (vector) notation by a single (matrix) index, as in SU(2) or SU(3) usage. We select an SU(4) basis (4 x 4, "v" matrices) in which the i = 1..8 correspond to setting the SU(3) Ai matrices in the upper left-hand corner of the v matrix with that index and define similar matrices for the rest. Since we are dealing with SL( 4, R) rather than

44

SU(4), we have to multiply the real matrices by A, thus making these generators noncompact. With ai denoting the Pauli matrices, and [aih,2 denoting a al matrix placed in the [1,2] rows and columns of the v matrix, we have a basis, Using the definition of the fijk (totally antisymmetric) and dijk (totally symmetric) coefficients of su(3), generalized to su(4) and corrected by the factors A for the symmetric matrices in the su(4) basis when changing to sl(4, R) as indicated above, we get coefficients !ijk and d ijk whose symmetry properties are thus reduced to the first two indices only. We can now write the Lie superbrackets as, [Qi,Qj] [[Qf,Nt] [[Qf,Nt] [[Qr,Nt] [[Q;,Nt] [[Qr,Nn [[Qf,Nn {Nt, N j-} {Nt,Ni- }

4

= = = = =

0

=

0

=

2dijkQk 2iQf

2i!ijkQk + 2zfijkNk .A

+

A

2dijk N k 2iN:t• 2idijk N;;

(3)

A

A

The Superconnection, Supercurvature and the Lagrangian

At this stage we set up the relevant superconnection a la Quillen, as an ad hoc algorithm (we shall later discuss the possibility of generating it from the matter fields' fiber bundle, by using a Connes-Lott type of product base space). The superconnection will thus be given as

(4) The nonvanishing v.e.v. field 1>(x) = ~t will occupy the main diagonal of quadrant II. This will also be the structure of the matrix derivative &, &_

-

(00

i.1 4x4

0

)

(5)

The resulting (generalized) curvature is then, (6)

where ~ +, ~ _ respectively denote the symmetric (in quadrant II) and antisymmetric (in quadrant III) components of ~ H. The first two terms arise for the 15 ilP, the last three appear for the 16 RH. In addition to its action on the Grassmann algebra -replacing an n-form by a (4-n)-form - the * duality operator conjugates the supermatrix. The R /\ * R gauge Lagrangian will thus consist of the following terms (a) R/\ *R, the SKY Lagrangian 21 - 23 ,

45

(b) 1-1 2 tP2 , the - mass term, once < olcilO >=/; O. (c) I{ -, +}12, the quartic Higgs potential V4 • (d) (D+)2 the + kinetic energy and gauge interaction, (e) (D _ )2, the _ kinetic energy and gauge interaction, (f) 18-1 2, the "negative squared mass" term V2 , triggering the spontaneous breakdown of local G symmetry, through 8hl~+r4) = 0 (g) There is no {+, ci} term, so that the 9 traceless components of + do not acquire mass. Moreover, they become the longitudinal G/ F = 8L(4, R)/80(4)gauging components of the connection, which acquire mass under the spontaneous symmetry breakdown.

5

A Connes-Lott Like Geometry for Matter

We now discuss a Connes-Lott like derivation. We stick to the chiral Z2 grading, i.e. to the product space Z2 . -2/3 hdt 2 + >.1/3

(d~2 + r2 dn~)

,

where

(4) 2

h = 1- ro

r2

rr

= r5sinh20 ,

r~

= r5sinh2'Y ,

r~

= r5sinh2O'

(5) (6)

This is just the five-dimensional Schwarzschild metric with the time and space components rescaled by different powers of >.. The event horizon is at r = ro. Several thermodynamic quantities can be associated to this solution. They can be computed in either the ten dimensional or five dimensional metrics and yield the same answer. For example, the ADM energy is

M

=

RVr2

29 20 (cosh 20 + cosh 2'Y + cosh 20') .

(7)

The Bekenstein-Hawking entropy is S

AlO

A5

= 4G]J = 4G~ =

27rRVr~

92

cosh 0 cosh 'Y cosh O'.

(8)

where A is the area of the horizon and we have used that the Newton constant is G]J = 871'6 9 2 • The Hawking temperature is T=

1 27rro cosh 0 cosh 'Y cosh a

(9)

58

The extremal limit corresponds to TO ~ 0, a, ,,(, (7 finite. In that limit the entropy (8) becomes 16,17

~ 00

keeping the charges (3)

(10)

and the temperature vanishes. Note that the extremal entropy is independent of any continuous parameters 22,23. The extremal black hole backgrounds preserve some space time supersymmetries and therefore they are BPS states. In this case the cosmic censorship bound becomes identical to the supersymmetry BPS bound 24

The near extremal limit corresponds to TO small and a, ,,(, (7 large. The relative values of a, ,,(, (7 are related to the total contribution of the different charges to the mass (7). The near extremal black holes that are easiest to analyze in terms of D-branes are those where (7 «a,,,(, or TO,T n «TI,T5, which means that the contribution to the mass (7) due to the D-branes is much bigger than the contribution due to the momentum excitations. This limit is called "dilute gas" 25 26. In this limit, the mass and entropy of the near extremal black hole become M

= Q5 RV + Q1R + RVT~ cosh 2(7 , 9

8

9

2g2

= 211" Rv'VTo V~ Q1Q5 cosh (7 .

(11)

(12)

9

Note that the five dimensional Reissner-Nordstrl1lm solution corresponds to the case of a = "( = (7 which is not included in the dilute gas limit. All these black hole solutions will be well defined if curvatures are everywhere much smaller than a' , since otherwise a' corrections to the low energy action become important. This generically implies that the sizes of the black hole should obey T1,T5,T n » 1 (remember that a' = 1). The precise condition will be a little more complicated if some scale is very different from the others. If the compactification sizes are of the order of a' this implies that gQ1 »1,

gQ5» 1,

g2N» 1

(13)

Note that we cannot enforce (13) by setting Q = 1 and making g» 1 since the condition (13) was derived in weakly coupled string theory, if 9 » 1 the corresponding bound comes from light D-strings and implies again that Q » 1. Actually, in the S-dual picture we obtain a black hole carrying NS fivebrane charge. In that case the black hole condition is just Q > > 1. Therefore black holes always involve large values of the charges. 3

D-brane description of extremal 5d black holes

We continue with type lIB string theory on T 5 = T4 X 8 1 • We consider a configuration of Q5 D-fivebranes wrapping the whole T5, Q1 D-strings wrapping the 8 1 and momentum N / R9 along the 8 1 , choosing this 8 1 to be in the direction 9. All charges N, Q1, Q5 are integers. For a review of D-branes see 27. We take the

59

coupling constant 9 to be small and the radius R9 to be large. The total mass of the system is M = QsRV + Q1R + N (14) 9

9

R

and it saturates the corresponding BPS bound. We will calculate the entropy of this state in perturbative string theory. This calculation was first done by Strominger and Vafa 10. We will describe three equivalent ways of doing the calculation, they are related by duality. We present it in three ways to emphasize the different aspects. All three use properties of D-branes.

3.1

Description in terms of open strings

This picture was proposed in 30,31,20. Since extremal D-branes are boost invariant along the directions parallel to the branes they cannot carry momentum along 8 1 by just moving rigidly. Our first task will be to identify the D-brane excitations that carry the momentum. The BPS mass formula for the whole system implies that these excitations have to be massless and moving along the 8 1 since the excitation energy, defined as the total mass of the system minus the mass of the onebranes and fivebranes, is equal to the momentum. If any excitation fails to be massless it would contribute more to the energy than to the momentum and the BPS mass formula would be violated. Excitations of the branes are described by open strings. There are many types of open strings to consider: those that go from one I-brane to another I-brane, which we denote as (1,1) strings, as well as the corresponding (5,5), (1,5) and (5,1) strings (the last two being different because the strings are oriented). Each of these types of strings have some massless and some massive modes, we will concentrate on the massless modes. The fact that the (1,5) strings have massless modes is non-trivial and it is basically due to the fact that DI-branes and D5-branes preserve some common supersymmetries. It will be important that the (1,5) strings have 4 bosonic and 4 fermionic massless modes. Actually, in our case we have 4Q1 Qs massless modes since the strings can go from any DI-brane to any D5-brane. The (1,1), (5,5) and (1,5) strings interact among themselves. We are interested in the low energy limit of these interactions. This corresponds to the field theory limit of the system of branes. If we take the size of the Sl along 9 to be very large this will be a 1+1 dimensional gauge theory. The Lagrangian is then determined by supersymmetry and gauge invariance. The (1,1) and (5,5) strings are U(Q1) and U(Qs) gauge bosons respectively and the (1,5) strings (and (5,1) strings) are in the fundamental (antifundamental) OfU(Q1)XU(Qsf. This Lagrangian was studied by Douglas 28. It turns out that after these interactions are taken into account only 4Q1 Qs truly massless degrees of freedom remain. In gauge theory terms, one is interested in the Higgs branch of the theory which is 4Q1 Qs-dimensional. We will spare the details which can be found in 20 (page 52), and conclude that the number of massless states is 4Q1 Qs (with the same number of bosonic and fermionic aThe gauge theory is in reality a bit more complicated since we are on a torus and one has to include the multiple "images" of the Dl-branes. Notice however that most of the extra states are highly massive, they are only necessary to ensure the right periodicity conditions on the torus.

60

states since the theory on the branes is supersymmetric). These massless degrees of freedom are described by a two dimensional conformal field theory 10, which accounts for the low energy excitations of the D-brane system. In fact, it is a (4,4) superconformal field theory, i.e. it has four left moving and four right moving supersymmetry generators. The rotational symmetry SO(4h234 '" SU(2)LX SU(2)R of the four spatial extended dimensions, that are transverse to the black hole, acts on this superconformal field theory as the SU(2)LXSU(2)R R-symmetries of the N=4 supersymmetry algebra in two dimensions. Notice that the chirality in space becomes correlated with the chirality of the 1+1 dimensional theory. So that SU(2)L in spacetime acts on the left movers and SU(2)R on the right movers 29 .

D-fivebranes

Momentum carried by the open strings.

9j Compact directions

--»

6

FIGURE 1: Configuration of intersecting D-branes. Open strings go between different branes The BPS state that we are interested in has only left moving excitations so the rightmovers are in their ground state ER = 0, The state counting is the same as that of a one dimensional gas of left moving particles with NB,F = 4Q1Q5 bosonic and fermionic species with energy E = EL = N I Rg on a compact one dimensional space of length L = 27r R 9 • The standard entropy formula gives 10 30

(15) in perfect agreement with (10), including the numerical coefficient. It might seem surprising that a system could have entropy at zero temperature, but this is a common phenomenon. Consider for example a gas of massless particles in a box with periodic boundary conditions constrained to have a fixed amount of momentum, at T = 0 the entropy remains nonzero. This reason is exactly the same reason that black hole entropy is nonzero at T = O. In our previous argument we implicitly took the D-strings and the fivebranes to be singly wound since we were assuming that the excitations carried momentum quantized in units of liR. For large N, N » Q1Q5, the entropy (15) is the same no matter how the branes are wound. However for N '" Q1 Q5 the winding starts

61

to matter. The reason is that in order for the asymptotic entropy formula to be correct for low N we need to have enough states with small energies 31. Let us study the effect of different wrappings. We first simplify the problem and consider a set of Q1 1-branes wrapped on 8 1 , ignoring for the time being, the 5-branes. We may distinguish the various ways the branes interconnect. For example, they may connect up so as to form one long brane of total length R' = RQ1. At the opposite extreme they might form Q1 disconnected loops. The spectra of open strings is different in each case. For the latter case the open strings behave like Q1 species of 1 dimensional particles, each with energy spectrum given by integer multiples of 1/ R. In the former case they behave more like a single species of 1 dimensional particle living on a space of length Q1 R. The result 32 is a spectrum of single particle In other words the system simulates a energies given by integer multiples of spectrum of fractional charges. For consistency the total charge must add up to an integer multiple of 1/ R but it can do so by adding up fractional charges.

Q!R .

Now let us return to the case of both 1 and 5 branes. By suppressing reference to the four compact directions orthogonal to x 9 we may think of the 5 branes as another kind of 1 brane wrapped on 8 1 . The 5-branes may also be connected to form a single multiply wound brane or several singly wound branes. Let us consider the spectrum of (1,5) type strings (strings which connect a 1-brane to a five-brane) when both the 1 and 5 branes each form a single long brane. The 1-brane has total length Q1R and the 5-brane has length Q5R. A given open string can be indexed by a pair of indices [i,]] labeling which loop of 1-brane and 5-brane it ends on. As a simple example choose Q1 = 2 and Q5 = 3. Now start with the [1,1] string which connects the first loop of 1-brane to the first loop of 5-brane. Let us transport this string around the 8 1 . When it comes back to the starting point it is a [2,2] string. Thansport it again and it becomes a [1,3] string. It must be cycled 6 times before returning to the [1,1] configuration. It follows that such a string has a spectrum of a single species living on a circle of size 6R. More generally, if Q1 and Q5 are relatively prime the system simulates a single species on a circle of size Q1 Q5R. If the Q's are not relatively prime the situation is slightly more complicated but the result is the same. A more detailed picture of how this happens is presented in 33. We can easily see that this way of wrapping the branes gives the correct value for the extremal entropy. As above, the open strings have 4 bosonic and 4 fermionic degrees of freedom and carry total momentum N / R. This time the quantization length is R' = Q1Q5R and the momentum is quantized in units of (Q1Q5R)-1. Thus instead of being at level N the system is at level N' = NQ1Q5. In place of the original Q1 Q5 species we now have a single species. The result is (16)

So we have a long effective string that is moving along the fivebrane. In the extremal case, this effective string picture follows precisely from an analysis of the moduli spaces of BPS states 34. What one actually has is a sum over multiple string states which one can call "second quantized" strings on a fivebrane 35. The state in which they are all connected into a single long string is the one having most entropy.

62

9.2

Description in terms of instantons

We now present the same calculation but in a picture where we start with just D-fivebranes and we build up the charges as excitations of the D-fivebranes. We start with Q5 D-fivebranes. The low energy theory on the fivebranes is an U(Q5) supersymmetric Yang Mills with 16 supersymmetries (same amount as N=4 in d=4). This theory contains BPS string solitons which are constructed as follows: if the fivebranes are along the directions 56789, take an instanton configuration that involves the directions 5678 and the gauge fields along those directions. The corresponding field configuration could be localized in the directions 5678 but will be extended along the direction 9, so it is a string soliton. Notice that, even though we call this solution an "instanton" (in the sense that the Yang-Mills fields are self dual solutions of a YM theory in four Euclidean dimensions (5678)), the physical interpretation is that we have a string "soliton" which exists for all times. It turns out that each instanton that lives on the fivebrane world volume carries one unit of D-string charge 28 due to a Chern Simon coupling on the fivebrane of the form (17) since F 1\ F will be proportional to the instanton number. We are interested in the case that the instanton number is Q1. SO D-strings dissolve into instantons when they get into fivebranes. In fact, giving an expectation value to the (1,5) strings corresponds to giving a size to the instanton 28,36. This description makes sense, in principle, only when the mass of the fivebranes is much bigger than the mass of the D-l-branes in (14), since otherwise the fivebranes would contain so much energy that they would no longer be described by the low energy YM theory. This instanton configuration is characterized by 4Q1 Q5 continuous parameters which specify the instanton positions on the branes as well as their relative orientation inside the U(Q5) gauge group, the space of these parameters is called "moduli space" . When we put some momentum along the direction 9, this momentum can be carried by small oscillations of the instanton configuration. We denote the instanton parameters by ~a, a = 1, .. ,4Q1Q5. They can be slowly varying functions ~a(t - x 9 ) representing traveling waves moving along the instantons. These are small oscillations in the parameters specifying the instanton configuration, i.e. oscillations in moduli space. Each bosonic mode has a fermionic superpartner and together they form a (4,4) superconformal field theory with central charge c = 6Q1Q5. In principle we can have a nontrivial metric on the moduli space of instantons. This metric will be hyperkahler, by supersymmetry. This implies that we indeed have a conformal field theory and furthermore, the central charge is given in terms of the dimension of the hyperkahler space without further corrections. A configuration carrying momentum N corresponds to states in the SCFT with Lo = N and £0 = 0, the entropy of such states can be calculated using the CFT formula d(N) ,..., e21rVNc/6 for the degeneracy of states at level N. This yields (15) again. The moduli space of instantons is, topologically, a symmetric product: M = (T4)QIQS/S(Q1Q5) 37,38. This is the target space of the SCFT, in other words: ~a(t, x 9) defines a map from R x S1 to M. Since we have twisted

63

sectors there are low lying modes with energies of the order of 1/RQl Qs which give rise to the long effective string picture described above. This picture where we start with only one kind of branes is the one that we would naturally use 39,40 if we are working in the M(atrix) theory Of4l. We should also address the question of whether the system really forms a bound state, this is a question on the behavior of the zero modes on the moduli space. The analysis of lO shows that they indeed form a bound state. Note that also for entropic reasons the state would stay together for a long time. It is interesting that when the momentum is not uniformly distributed along the string (the 9 direction) the D-brane calculation still agrees with the corresponding black hole result 43.

9.9 A more geometric picture We now consider aT-dual configuration which has two sets of D3-branes intersecting along a line. We have Q3 D3-branes oriented along the directions 569 and Q~ oriented along 789. We also have, as above, momentum N along the direction 9. This configuration is related by T-duality along 56 to the one considered above. Along the directions 5678 the configuration looks like two sets of two dimensional planes intersecting at points. We imagine the planes to be separated from each other. According to what we saw above, we expect that the degrees of freedom come from 33' strings. Exciting these massless strings actually corresponds to deforming the two intersecting planes into a smooth Riemann surface. In order to state this more precisely, let us consider a pair of intersecting D3- and D3'-branes. We will consider what happens locally at the intersection point. Let us define z = X s + iX 6 and w = X7 +iX8. We concentrate on what is happening in the directions 5678, the configuration will be translationally invariant (for the moment) along the direction 9. The two intersecting branes are described by the equation zw = 0, in other words, the brane is sitting at the points (z, w) which satisfy the above equation, which are, of course the plane w = 0 (one D3-brane) and z = 0 (the other D3'-brane). Exciting the 33' strings corresponds to deforming the shape of the threebranes so that they obey the equation zw = c where the constant c is related to the expectation value of the 33' massless strings. We now have a single smooth D3-brane which is wrapping once around each of the planes 56 and 78. A nonzero value for c smoothes out or "blows up" the singular intersection of the D3-branes. The three-branes will preserve some supersymmetries as long as the equation characterizing their shape is holomorphic 42. Since we have Q3Q~ intersection points we can deform each of them in this way. We end up with a single smooth D3 brane that is wrapping Q3 times the plane 56 and Q~ times the plane 78. This gives 2Q3Q~ real parameters characterizing the shape of the surface. There are other 2Q3Q~ parameters coming from Wilson lines of the U(I) gauge field that lives on this single D3 brane. If we look at each intersection point locally, once the brane is obeying the equation zw = c f 0 we see that we can have a gauge field A z = biz. Note that the degrees of freedom come mostly from deforming these intersection points, the global positions of the Q3 + Q~ branes is a subleading effect in Q. Actually, the counting of degrees of freedom can be done precisely 37,38 and the result is that we have

64

precisely 4Q3Q 3 bosonic degrees of freedom. By supersymmetry we have an equal number of fermionic degrees of freedom. They arise from the fermion that lives on the D3-brane. Since the D3-brane has a complicated shape, this fermion field will have 4Q3Q 3 zero modes. Now the momentum along 9 will be carried by oscillations of these massless degrees of freedom and the entropy is calculated as above, once we know the number of massless degrees of freedom. Notice that in this picture we are implicitly assuming that the radii of the compact dimensions are all very big. 3.4

Justification of the BPS counting

In performing these calculations we have assumed that the coupling was weak. One might naively think that the only condition is g « 1. However, since there is a large number of branes there could be large N / (= Q) effects which grow as gQ. At low energies these are just the large N effects of the Yang-Mills theory (note that the YM coupling is gYM = ,;0). In string theory they correspond to the possibility of inserting a hole on the worldsheet, each hole has a power of g and a factor of Q coming from the trace over Chan Paton factors. So the above calculations are correct when gQ« 1 (18) On the other hand the classical black hole solution is well defined when gQ » 1 13. The reason that we expect agreement (and find it) is that we are counting the number of BPS bound states and this number is not expected to change as we vary the coupling constant. Notice that indeed the extremal black hole entropy (10) is independent of the coupling and all other continuous parameters. Similarly, we assumed that the compactification radii were large, we can also argue that the number of BPS states will not change when we change the radii. 4

N ear extremal black holes

We now turn to a discussion of near extremal five-dimensional black holes (4) (11)(12). For the reasons that we have just discussed we might naively not expect agreement in this case. However, in the dilute gas regime, we are very close to a configuration of extremal Dl and D5 branes and supersymmetry non-renormalization arguments do indeed help us 44,45 and explain the agreement that we are going to find. We are going to consider a weakly coupled system of D-branes but we will always restrict to the low energy approximation. We will see that for the nearextremal case the agreement between the two approaches is just as impressive as in the extremal case. The D-brane model is a low energy approximation to the full quantum dynamics of black holes. The energy should be low compared to the scale set by gravitational size of the black hole Ts defined as the radius at which the redshift of a static observer becomes of order one, r; '" gQ '" r~ , r~. The condition on the energy becomes WT s « 1, where W represents the typical energy of the brane excitations, as well as the Hawking temperature of the system 44. We will not go into the details of the justification of this extrapolation which can be found in 44 .

65

We start with a system of ID-branes and 5-D branes as before and we add some extra energy and momentum to the system. This energy excites the massless left and right moving modes of the instanton configuration. In the instanton picture we say that we are creating left and right moving excitations on the moduli space, (t, x 9 ). Using the black hole formulas (3),(11) we calculate the energies of the left and right movers

e

M _ Q5 RV _ Q1R 9

N =

9

= RVr5 cosh 20" = NL + NR 2g

R 2Vr2 2g 2 0 sinh 20" = N L - N R .

R

,

(19)

(20)

In this fashion we can calculate N L,R in terms of the black hole parameters. The entropy calculation proceeds as in the extremal case. We work in the multiply wound picture with 4 bosons and 4 fermions with effective Ni,R = Q1 Q5NL,R. We find that

We see that this result agrees with the near extremal entropy in the dilute gas limit (12) once we use (19) (20). This is the simplest case, if we want to consider more general near extremal black holes, including Reissner-Nordstn!lm, one has to include other excitations besides the right movers 30,46 and the arguments are not so well justified. These non-BPS states will decay. The simplest decay process is a collision of a right moving excitation with a left moving one to give a closed string mode that leaves the brane. We will calculate the emission rate for uncharged particles. The basic process is a right moving mode with momentum P9 = n/RgQ1Q5 colliding with a left moving one of momentum pg = -n/R9Q1Q5 to give a closed string mode of energy w = 2n/RQ1Q5. Notice that we are considering the branes to be multiply wound since that is the configuration that had the highest entropy. If the momenta are not exactly opposite the outgoing string carries some momentum in the 9th direction and we get a charged particle from the five dimensional point of view. Notice that the momentum in the 9th direction of the outgoing particle has to be quantized in units of 1/ R 9 , only particles on the branes can have fractional momenta!. This means that outgoing charged particles have a very large mass, and that they are thermally suppressed when R9 is small. All charged particles have masses of at least the compactification scale. In other words, we have a very long effective string winding around the compact direction 9, it can oscillate along the other 4 compact dimensions (5678) and it emits gravitational quadrupole radiation. The graviton is polarized along the compact directions and is a scalar from the point of view of the five dimensional observer. We will calculate the rate for this process according to the usual rules of relativistic quantum mechanics and show that the radiation has a thermal spectrum if we do not know the initial microscopic state of the black hole.

66

----7

0

Hawking Radiation

Closed string massless modes

V

Compact dimensions

Extended dimension

IGURE 2: D-brane picture of the Hawking radiation emission process. The state of the D-branes is specified by giving the left and right moving occupation numbers of each of the bosonic and fermionic oscillators. In fact, the near extremal D-branes live in a subsector of the total Hilbert space that is isomorphic to the Hilbert space of a 1+1 dimensional eFT. The initial state IWi) can emit a closed string mode and become Iw f). The rate, averaged over initial states and summed over final states (as one would do for calculating the decay rate of an unpolarized atom) is (22) We have included the factor due to the compactified volume RV. The relevant string amplitude for this process is given by a correlation function on the disc with two insertions on the boundary, corresponding to the two open string states and an insertion in the interior, corresponding to the closed string state. We consider the case when the outgoing closed string is a spin zero boson in five dimensions, so that it corresponds to the dilaton, the internal metric, internal B,..v fields, or internal components of RR gauge fields. This disc amplitude, call it A, is proportional to the string coupling constant 9 and to w2 4 7 . The reason for this last fact is that it has to vanish when we go to zero momentum, otherwise it would indicate that there is a mass term for the open strings (since one can vary the vacuum expectation value of the corresponding closed string fields continuously). In conclusion, up to numerical factors, (23) Note that performing the average over initial and summing over final states will just produce a factor of the form p.dn)PR(n) with (24)

67 where Ni is the total number of initial states and a;: is the creation operator for one of the 4 bosonic open string states. The factor pL(n) is similar. Since we are just averaging over all possible initial states with given value of N R, this corresponds to taking the expectation value of a~an in the microcanonical ensemble with total energy ER = NR/ R9 = Nk/ R9Ql Q5 of a one dimensional gas. Because Nk is large compared to one, we can calculate (24) in the canonical ensemble. The occupation number is then

PR(W) =

e~ 1-

'"

e 2TR

We can read off the "right moving" temperature

(25) There is a similar factor for the left movers PL with a similar looking temperature

(26) In fact it would be more accurate to say that there is only one physical temperature of the gas, which agrees with the Hawking temperature of the corresponding black hole, 1 1 1 1 (27) = -(-+-) TH 2 TL TR and that Ti,k = T.ifl(l ± J.l) are some natural combination of the temperature and the chemical potential, which gives the gas some net momentum. Using the values for NL,R from (19) (20) we find

(28) The expression for the rate is, up to a numerical constant, d4 k

dr....., -

W

1 R LRVIAI2QIQ5RpR(W)PL(W)

(29)

PoPo

where A is the disc diagram result. The factor QIQ5R is a volume factor, which arises from the delta function of momenta in (22) Ln c5(w - 2n/ RQl Q5) ....., RQl Q5. The final expression for the rate is, using (23) in (29), 2

df _ 71"3 9 Q Q

-

V

1

5

W

1 '"

e2TL

1 -

'" 1 e2TR

-

d4 k 1 ()4 271"

(30)

We have not shown here how to calculate the precise numerical constant in front of (30) , this precise calculation was done in 48, and we refer the reader to it for the details.

68

If we are considering a black hole which is very close to extremality with nonzero momentum N '" NL » NR then we find from (26)(25) that TL »TR. Examining the expression for the rate (30) we see that the typical emitted energies are of the order of T R . Therefore, we can approximate the left moving thermal factor by 2TL

PL'" -

(31)

W

and replacing it in (30) we find elI'

=

27r 2 g 2 1 d4 k RV JQIQ5 N e ~ (2)4 2TR - 1 7r

= AH e

1 '" 2TR

-

d4 k (2 1 7r )4

(32)

where AH is the area of the horizon. We conclude that the emission is thermal, with a physical Hawking temperature (33) which exactly matches the classical result (9). The area appeared correctly in (30)49 . Actually, the coupling constant coming from the string amplitude A is hidden in the expression for the area (area = 4G~S). The overall coefficient in (32) matches precisely with the semiclassical result 48. Notice that if we were emitting a spacetime fermion then the left moving mode could be a boson and the right moving mode a fermion, this produces the correct thermal factor for a spacetime fermion. The opposite possibility gives a much lower rate, since we do not have the enhancement due to the large PL (31). When separation from extremality is very small, then the number of right movers is small and the statistical arguments used to derive (30) fail. Semiclassically this should happen when the temperature is so low that the emission of one quantum at temperature T causes the temperature to change by an amount of order T. This means that the specific heat is of order one. This happens when the mass difference from extremality is 50 (34) for a Reissner-Nordstr0m black hole, with re being the Schwarzschild radius of the solution. The D-brane approach suggests the existence of a mass gap 2

t5Mmin '" Ql Q5 R

(35)

which using (3) scales like (34). This is an extremely small energy for a macroscopic extremal black hole. In fact, it is of the order of the kinetic energy that the black hole would have, due to the uncertainty principle, if we want to measure its position with an accuracy of the order of its typical gravitational radius rs: 15M '" (tl.p) 2 1M with tl.p'" 1/rs. Now we calculate the entropy of a rotating black hole in five dimensions 29,51. The angular momentum is characterized by the eigenvalues on two orthogonal twoplanes, J1 , J2 , for example J 1 corresponds to rotations of the 12 plane and J2 to

69

rotations of the 34 plane. In terms of the J3 eigenvalues h,JR of the SU(2)LX SU(2)R ...... SO(4) decomposition of the spatial rotation group we find (36)

As we mentioned above J R , h are carried by right and left movers respectively. They are also the eigenvalues of U(I) appearing in the supersymmetry algebra. States carrying U(I) eigenvalue J have conformal weight bigger than Ll = 6J2 Ie where e = 6Ql Q5 is the central charge. The states with minimum conformal weight correspond to states eiJ¢IO}, where j = lc20rp is the U(I) current. In the total left moving energy NL there is an amount JJjQIQ5 which we are not free to distribute. It is fixed by the condition that the system has angular momentum JL, so the effective number of left movers that we are free to vary is fh = NL - JIIQIQ5. The same is true for the right movers, so that the entropy becomes

which agrees with the classical entropy formula of a rotating black hole in the dilute gas regime 51. Actually, in five dimensions we can have rotating BPS black holes by setting NR = JR = 0, this implies J l = J2. Again the corresponding formula agrees with the classical entropy formula but the restriction to the dilute gas regime is no longer necessary since the computation is protected by supersymmetry. 5

Greybody factors

The D-brane emission rate into massless scalars is given by (30). More precisely that is the emission into minimally coupled scalars, scalars that in the supergravity theory are not coupled to the vector fields that are excited in the black hole background. In other words, the action for these scalars is

where g is the Einstein metric in five spacetime dimensions. According to the semiclassical analysis the emission rate should be

df=a(w,ro,a,1,a)

1 w

eTH -1

(39)

where a(w,ro,a,1,a) is the absorption cross section of the black hole which is a function of the various parameters specifying the black hole solution (4). In the usual Schwarzschild black hole case the only scale in the solution would be the Schwarzschild radius rs. This emission rate (39) has the same form for any body emitting thermal radiation. The absorption cross section comes in because of detailed balance: in order for the body to be in equilibrium with a bath of radiation it has to absorb as much as it emits. The prefactor in (39) is usually called greybody factor, since it is what makes bodies grey instead of black. At first sight, the semiclassical rate (39) does not seem to be in agreement with the

70 D-brane rate (30) since one has two exponential factors and the the other seems to have only one. In order to see whether they really agree we should calculate the greybody factor. It turns out that the greybody factor is precisely such that these two calculations to agree. We now describe this calculation. We consider the scattering of scalars from a five dimensional black hole in the dilute gas limit ro, r n « rl, rs. We also restrict to low energies satisfying w « l/rl, l/rs but there is no restriction on TITe; wrg or wrn , in other words, no TITe; restriction on w/TL , W/TR' We follow the notation of 26 , where further details of the geometry may also be found. The wave equation in this background becomes

hd difJ 2 3-d r 3h- +w >'ifJ = 0, r r dr

(40)

where >., h are defined in (5). We divide space into a far region r » rI, rs and a near region r « l/w and we will match the solutions in the overlapping region. In the far region, the equation is solved by the Bessel functions 1 ifJ = - [aJv(p) , p

,

+ ,8Lv(p)],

(41)

with p = wr and v 2 = 1 - €, where € = w2(r~ + r~) is very small and we keep it to simplify the form of the intermediate equations but will disappear from the final answer. From the large p behavior the incoming flux is found to be

fin

= Im(ifJ*r30rifJ) = _1_lae iV1T / 2 + ,8e-iv1T/212.

(42)

211'W2

On the other hand, the small p behavior of the far region solution is

Now we turn to the solution in the near region r near region wave equation is

difJ (1 - v) 2~ifJ - 2 - (1 - v)dv

dv

«

l/w. Defining v

D E)

+ ( C + - + - 2 ifJ = V

v

= r5;r2, the

0

(44)

where C

v 2 -1 E=---

= (Wrnrlrs)2 2r5

4

(45)

Defining (46) with A a constant, we find that the solution to (44) with only ingoing flux at the horizon is given by (46) with the hypergeometric function

F = F(a,b,c; 1- v)

(47)

71

. w a=-v/2+1/2+i : , b = -v/2 + 1/2 + i 47r';R ' c = 1 + z27rT 4 TL H The behavior for small v can be calculated by expressing the hypergeometric function (47), which depends on 1 - v, in terms of hypergeometric functions depending on v and then expanding in v. Matching this with (43) we find

a

/2 = A

r(1

+ i_W_) 21rTH

1

[ r(1 + i 41rTL )r(1 + i 41rTR )

(48)

,

The absorbed flux is 2

Jabs

wro 1A 12 . = Im(cP * hr3 8r cP) = 27rTH

(49)

The absorption cross section for the radial problem is given by the ratio of the two fluxes (49) (42). The plane wave cross section is obtained by multiplying by 47r/w 3

e~ -1

(50)

where the exponential terms come from the gamma functions in (48). We see that it has precisely the right form to make the D-brane result (30) agree with the semiclassical calculation (39). Notice that we are doing two very different calculations. On one hand we are considering a quantum field propagating on a fixed classical supergravity background and on the other we are doing a computation in flat space with some Dbranes at the origin. This might seem surprising at first but it is not once we realize that the low energy condition wr9 « 1 implies that the black hole looks like a point for such long wavelengths. Actually the most surprising and interesting fact is that in the D-brane calculation we are computing a thermal rate, agreement is found only after we average over the states of the D-branes. This suggests that the classical gravity approximation is a thermodynamic approximation to quantum gravity. These greybody factor calculations have been generalized to various cases. One possible generalization is to consider the emission of scalars that are not minimally coupled, in some cases precise agreement is found 52 53. From the D-brane point of view the difference between these scalars and the one that we have been considering is in the conformal weight of the operator on the effective SCFT that they couple to. The minimally coupled scalars couple to operators of dimension (1,1) (like 8X8X) while the scalars in 52 couple to operators of conformal weights (2,2) or (1,2) in 53. There is still some puzzling disagreement for the case of operators of weight (3,1) 54, which hopefully will be resolved soon!. Recently some greybody factors for fermions and vectors were calculated and found to agree with the D-brane model 58 . Another generalization is to consider the emission of higher partial waves 55,57,56. These calculations of grey body factors shows that some of the features of the near extremal geometry are encoded in the dynamics of the 1+1 dimensional gas (or the CFT). Since the wavelengths of the particles we scatter are much bigger than the

72

size of the black holes it is hard to get precise information about the metric. A more direct way to obtain information about the metric is by using D-brane probes 59. In that approach one starts from D-branes in flat space and by integrating out the massive stretched open strings one obtains an action for the probe D-brane that is at some distance from the rest of the branes. This action is then interpreted as the action of a D-brane in the presence of some classical supergravity background. This works to one loop 59,60 but the status of the higher loop contributions is unclear. 6

Final remarks

We have seen how string theory, using D-branes provides a microscopic understanding of the black hole entropy. It is possible to study extremal and near extremal black holes in some particular limit ("dilute gas"). The description not only accounts for the entropy but also captures the full low energy dynamics of the black hole, such as low energy absorption and emission cross sections. Hawking radiation coming from these near extremal black holes can always be described within the low energy approximation. Since the D-brane model is unitary we can conclude that the low energy dynamics of these black holes is unitary. One might wonder whether there are higher energy processes that violate unitarity. Well, it could certainly be the case. Notice, however, that the information loss paradox for these black holes could be put just in terms of low energy processes: sending in some low energy waves and watching the black hole evaporate, this is a process that according to the semiclassical arguments would produce a large amount of entropy 64. The D-brane model, on the other hand predicts that very little entropy (if any) is created. We say very little because it is possible that the higher energy processes might be not unitary. Some reasons why the traditional semiclassical arguments would not apply for this case were explored in 65. The grey body factor calculation shows that many features of the geometry close to the horizon (close relative to the size of the black hole, but far compared to the Planck length) are reproduced by the gas of left and right movers. Complicated Smatrix elements for absorption and emission are reproduced. It is important that in order to reproduce them we have to do an average over the black hole microstates. This is saying that the classical gravity approximation itself is, in some sense, a thermodynamic approximation. Ideas along this line have been envisioned in 66. This is, in my opinion, the tip of the iceberg in a new understanding of quantum gravity and the relation of gauge theories to gravitation. In 2+ 1 dimensions the entropy was explained in terms of boundary degrees of freedom of a Chern Simons theory by Carlip 67, since 2+ 1 gravity can be shown to be equivalent to a Chern Simons theory. It would be interesting to understand the relation between that picture and the string theory picture. There is clear relation of this description of black holes and Matrix theory 41. In the D-brane description of black holes the spacetime physics close to the horizon is described in terms of some gauge theory living on D-branes. In Matrix theory the whole spacetime theory arises from the gauge theory living on branes. The Matrix description of black holes is just the D-brane description reinterpreting the D5-brane charge as momentum 40,39.

73 The four dimensional black holes have a similar description 61,62,63, and there is a similar "dilute gas" limit. It was actually shown in 68 that there is another dilute gas limit for the near extremal Kerr black hole in five dimensions which also has greybody factors suggesting a description in terms of a 1+1 dimensional gas. Acknowledgments I thank C. Callan, G. Horowitz, D. Lowe, A. Strominger, L. Susskind and A. Peet, for many fruitful collaborations on this subject. 1. S. Hawking and G. Ellis, The Large Scale Structure of Spacetime, Cambridge Univ. Press, 1973. 2. R. Penrose, Phys. Rev. Lett. 14 (1965) 57. R. Penrose, "Singularities and Time Asymmetry" and R. Geroch and G. Horowitz "Global Structure of Spacetimes", both in General Relativity, an Einstein Centenary Survey, ed. S. Hawking and W. Israel (Cambridge University Press)(1979) 3. R. Wald, Talk at the American Physical Society meeting, Washington, April 1996. 4. S. Hawking, Phys. Rev. Lett. 26 (1971)1344. 5. S.W. Hawking, Comm. Math. Phys. 43 (1975) 199. 6. J. Beckenstein, Phys. Rev. D7 (1973) 2333; Phys. Rev. D 9 (1974) 3292; S. W. Hawking, Phys. Rev. D13 (1976) 191. 7. For some reviews see: M. Green, J. Schwarz, and E. Witten, "Superstring Theory," two volumes (Cambridge University Press, 1987); A.M. Polyakov, "Gauge Fields and Strings," (Harwood, 1987); M. Kaku, "Introduction to Superstrings," (Springer-Verlag, 1988) ; D. Lust and S. Theisen, "Lectures on String Theory," (Springer-Verlag, 1989). 8. J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 9. hep-th/9510017;J. Dai, R. Leigh and J. Polchinski, Mod. Phys. Lett. A 4 (1989) 2073; P. Horava, Phys. Lett. B23l (1989) 251. 10. A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99, hep-th/9601029. 11. L. Susskind, RU-93-44, hep-th/9309145 12. A. Sen, Mod. Phys. Lett. AlO (1995) 2081, hep-th/9504147. 13. G. Horowitz and J. Polchinski, hep-th/9612146. 14. G. Horowitz, gr-qc/9704072; gr-qc/9604051. 15. C. Hull and P. Townsend, Nucl. Phys. B438 (1995) 109, hep-th/9410167. 16. A. Tseytlin, Mod. Phys. Lett. All (1996) 689, hep-th/9601177. 17. M. Cvetic, Review talk, hep-th/9701152 and references therein. 18. R. Khuri, hep-th/9704110. 19. G. Horowitz, J. Maldacena and A. Strominger, Phys. Lett. B383 (1996) 151, hep-th/9603109. 20. J. Maldacena, Ph.D. Thesis, Princeton University, hep-th/9607235 21. J. Maharana and J. Schwarz, Nucl.Phys. B390 (1993) 3; A. Sen, Nucl. Phys. D404 (1993) 109. 22. S. Ferrara, R. Kallosh and A. Strominger, Phys. Rev. D 52 (1995) 5412, hep-th/9508072; M. Cvetic and D. Youm, Phys. Rev. D53 (1996) 584, hepth/9507090; G. Gibbons and P. Townsend, Phys. Rev. Lett. 71 (1993) 3754. 23. F. Larsen and F. Wilczek, Phys. Lett. B375 (1996) 37, hep-th/9511064.

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24. R. Kallosh, A. Linde, T. Ortin, A. Peet, A. Van Proeyen, Phys. Rev. D46 (1992) 5278, hep-th/9205027. 25. G. Horowitz and A. Strominger, Phys. Rev. Lett. 77 (1996) 2368, hepth/9602051. 26. J. Maldacena and A. Strominger, Phys. Rev. D55(1997)861, hep-th/9609026. 27. J. Polchinski, S. Chaudhuri and C. Johnson, Notes on D-Branes, hepth/9602052j J. Polchinski, Tasi Lectures on D-branes, hep-th/9611050. 28. M. Douglas, hep-th/9512077. 29. J. Breckenridge, R. Myers, A. Peet and C. Vafa, Phys. Lett. B391 (1997) 93, hep-th/9602065. 30. C. Callan and J. Maldacena, Nucl. Phys. B472 (1996) 591, hep-th/9602043. 31. J. Maldacena and L. Susskind, Nucl. Phys. B475 (1996) 679, hepth/9604042. 32. S.R. Das and S.D. Mathur, Phys. Lett. B375 (1996) 103, hep-th/9601152. 33. S. Hassan and S. Wadia, hep-th/9703163. 34. R. Dijkgraaf, G. Moore, E. Verlinde and H. Verlinde, hep-th/9608096. 35. R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B486 (1996) 77, hep-th/9603126j Nucl. Phys. B486 (1996) 89, hep-th/9604055j Nucl. Phys. B484 (1996) 543, hep-th/9607026. 36. E. Witten, Nucl. Phys. 460 (1996) 541, hep-th/9511030. 37. C. Vafa, Nucl. Phys. B463 (1996) 415, hep-th/9511088j Nucl .Phys. B463 (1996) 435, hep-th/9512078 38. M. Bershadsky, C. Vafa and V. Sadov, Nucl. Phys. B463 (1996)398,hepth/9510225. 39. M. Li and E. Martinec, hep-th/9703211j hep-th/9704134. 40. R. Dijkgraaf, E. Verlinde and H. Verlinde, hep-th/9704018. 41. T. Banks, W. Fischler, S. H. Shenker, L. Susskind, Phys.Rev. D55 (1997) 5112, hep-th/9610043. 42. K. Becker, M. Becker and A. Strominger, Nucl. Phys. B456 (1995) 130, hep-th/9507158. 43. G. Horowitz and D. Marolf, Phys .Rev. D55 (1997) 835, hep-th/9605224j Phys. Rev. D55 (1997) 846, hep-th/9606113. 44. J. Maldacena, hep-th/9611125. 45. S. Das, hep-th/9703146. 46. J. Maldacena, Nucl. Phys. B477 (1996) 168, hep-th/9605016j 1. Klebanov and A. Tseytlin, Nucl. Phys. B475 (1996) 164, hep-th/9604166j V. Balasubramanian and F. Larsen, Nucl. Phys. B478 (1996) 199, hep-th/9604189j A. Hanany and 1. Klebanov, Nucl. Phys. B482 (1996) 105, hep-th/9606136j 1. Klebanov and A. Tseytlin, Nucl. Phys. B479 (1996) 319, hep-th/9607107 47. A. Hashimoto and 1. Klebanov, Phys. Lett. B381 (1996) 437, hepth/9604065j M. Garousi and R. Myers, Nucl .Phys. B475 (1996) 193, hepth/9603194. 48. S. Das and S. Mathur Nucl. Phys. B478 (1996) 561, hep-th/9606185j Nucl. Phys. B482 (1996) 153, hep-th/9607149. 49. A. Dhar, G. MandaI and S. Wadia Phys. Lett. B388 (1996) 51, hepth/9605234.

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50. J. Preskill, P. Schwarz, A. Shapere, S. Thivedi and F. Wilczek, Mod. Phys. Lett.A 6 (1991) 2353j C. Holzhey and F. Wilczek, Nucl. Phys. B380 (1992) 447, hep-thj9202014; P. Kraus and F. Wilczek, Nucl. Phys. B433 (1995) 403. 51. J. Breckenridge, D. Lowe, R. Myers, A. Peet, A. Strominger and C. Vafa, Phys. Lett. B381 (1996) 423, hep-thj9603078. 52. C. Callan, S. Gubser, 1. Klebanov, A. Tseytlin, Nucl. Phys. B489 65, hepthj9610172; 1. Klebanov and M. Krasnitz, Phys. Rev. D55 (1996) 3250, hep-thj9612051. 53. 1. Klebanov, A. Rajaraman, A. Tseytlin, hep-thj9704112. 54. M. Krasnitz, 1. Klebanov, hep-thj9703216. 55. J. Maldacena and A. Strominger, hep-thj9702015. 56. S. Mathur, hep-thj9704156. 57. S. Gubser, hep-thj9704195. 58. S. Gubser, hep-thj9706100. 59. M. Douglas, J. Polchinski and A. Strominger, hep-thj9703031. 60. J. Maldacena, hep-thj9705053. 61. J. Maldacena and A. Strominger, Phys. Rev. Lett. 77 (1996) 428, hepthj9603061; C. Johnson, R. Kuhri and R. Myers, Phys. Lett. B378 (1996) 78, hep-thj9603062. 62. G. Horowitz, D. Lowe and J. Maldacena, Phys. Rev. Lett. 77 (1996) 430, hep-thj9603195. 63. S. Gubser and 1. Klebanov, Phys. Rev. Lett. 77 (1996) 4491, hep-thj9609076. 64. S. Hawking, Phys. Rev. D14 (1976) 2460. 65. T. Jacobson, hep-thj9705017. 66. T. Jacobson, Phys .Rev. Lett. 75 (1995) 1260, gr-qcj9504004. 67. S. Carlip, Phys. Rev. D51 (1995) 632, gr-qcj9409052. 68. M. Cvetic and F. Larsen, contribution to SUSY97, hep-thj9708090j also hepthj9705192.

76 The notions of time and evolution in quantum cosmology R. Parent ani Laboratoire de Mathematiques et Physique Theorique, CNRS UPRES A 6089 Faculte des Sciences, Universite de Tours, 97200 Tours, France Abstract We re-examine the notions of time and evolution in the light of the mathematical properties of the solutions of the Wheeler-DeWitt equation which are revealed by an extended adiabatic treatment. The main advantage of this treatment is to organize the solutions in series that make explicit both the connections with the corresponding Schrodinger equation as well as the modifications introduced by the quantum character of gravity. When the universe is macroscopic, the ordered character of the expansion leads to connections with the Schrodinger equation so precise that the interpretation of the solutions of the Wheeler-DeWitt equation is unequivocally determined. On the contrary, when the expansion behaves quantum mechanically, i.e. in the presence of backscattering, major difficulties concerning the interpretation persist.

1

Introduction

It is now well known that in quantum cosmology, when working with the solutions of the Wheeler-DeWitt (WDW) equation, one completely looses the notion of time, i.e. the notion of an external parameter that clocks events. Indeed, the WDW constraint equation implies that the universe is in an eigenstate of zero total energy_5. In a Schrodinger sense, this means that one works with a stationary state. Before investigating the problem of the description of evolution in quantum cosmology, it should be noted that in classical cosmology, when working with the action in the Hamilton-Jacobi formalism, one also looses the notion of the (Newtonian) time. As in quantum cosmology, this results from the invariance of the theory under arbitrary reparametrizations of time. It should also be noticed that a similar disappearance applies to space as well, in virtue of the full reparametrization invariance of general relativity. Surprisingly, this problem has attracted much less attention that the one associated with time, see however 678. In any case, when analyzing the solutions of the Wheeler-DeWitt equation, one must keep in mind that certain properties might posses an equivalent version in classical cosmology and, more importantly, one should sort out these classical aspects from those which are specific to quantum gravity. These concerns are rarely found in the literature, see however 91m!. The absence of time in quantum cosmology leads at least to three questions: 1. Is the notion of time intrinsic to the notion of (quantum) evolution? 2. How evolution should be described in the absence of time? 3. When and why this new description coincides with the usual one based on a Schrodinger equation ? That time might not be necessary in cosmology is not really a surprise. For instance, cosmological events are delivered to astronomers ordered in terms of their red shifts z = areception/aemi88ion -1 which bear no direct information about lapses

77

of proper time At between emission and reception. Indeed, the determination of the latters requires the knowledge of the expansion rate a( a) = da/ dt from a em i88ion until areception. Thus, at first sight, since one has lost the notion of time, it seems that this rate is not physically meaningful. However, it becomes meaningful when it is compared to rates governing local processes, such as growth of perturbations. This is because these rates issue from evolution laws governed by lapses of (proper) time. This discussion brings about an important concept that we shall use throughout in this paper: it is the comparison of the rates characterizing two different processes that leads, a posteriori, to the notion of time. This is why we shall analyze the transitions amplitudes of heavy atoms induced by absorption or emission of photons that occur in an expanding universe. Notice that the physical necessity of comparing two rates to re-introduce the notion oftime is reminiscent of the description of planar orbits of an isolated system in Newtonian mechanics. In that case, one can describe intrinsically, i.e. without reference to an external time, the trajectory in the form r(8). However, to give physical sense to the "extrinsic" description r(t),8(t), one needs an additional system to tell what t meangl ffi1l2 • To further discuss the issues raised by the three questions, one must consider technical aspects more closely. Firstly, we shall pursue the discussion in minisuperspace. In this restricted space, classical cosmology is described by the history of the scale factor a(t). The main advantage of this truncation is that it leads to explicit equations whose solutions can be fully analyzed. The physical relevance of this radical truncation is greatly enlarged once it is conceived as the first order term in an expansion in local gravitational deformationSll. In particular, in such an expansion, it is consistent to consider quantum matter fields carrying non vanishing momenta evolving in homogeneous three geometries, thereby allowing the study of local phenomena'!. When mini-superspace is conceived in this manner, the rather symmetrical roles played by a and 4> (the homogeneous part of a matter field) in a strict mini-superspace reduction become clearly distinguishable since the local perturbations of all fields (matter and gravity) are minimally coupled to a and not to 4>. Secondly, of great importance, is the choice of the formalism used to investigate the solutions of the WDW equation. Indeed, the choice of the formalism plays a double role. First, it organizes the mathematical approximations necessary to construct non-trivial solutions. Secondly, it partially determines the physical interpretation of these solutions. In fact, a careful examination of the literature reveals that authors who use very different mathematical expansions reach generally different conclusions concerning the interpretative aspects, see 1_5,13. In any case, at present, no fully consistent interpretation of the solutions of the WDW equation has been reached. At the end of this paper, we shall present the reasons that prevent a simple interpretation of these solutions. However, in spite of these difficulties, unambiguous answers to the three questions raised above can be obtained in certain cases. To this end, we shall apply, following 14, an (extended) adiabatic treatment to the WDW equation. The merits of this treatment are the following. First, when applied both to the Schr6dinger and the WDW equations describing the same quantum processes, this treatment makes the comparison of both descrip-

78

tions of the transitions quite transparent. This is because, in both descriptions, the dynamical role of the expansion rate of the universe is made explicit. Indeed, nonadiabatic transitions are directly induced by the expansion. This is to be opposed to time dependent perturbation theory wherein transitions are induced by the interaction hamiltonian and bear therefore no direct relation to cosmology. Notice also that in the absence of adiabatic transitions, there is nothing to clock. Thus, time cannot be recovered from a universe in which matter is in an eigenstate (or a frozen superposition thereof). In fact, we shall recover the notion of time from the transition rates induced by the expansion. In this, the analysis of non-adiabatic transitions precisely formalizes the idea expressed above following which time reappears from a comparison of the rates of two physical processes. Indeed, no reference to external properties, like the trajectory of the "peak,a of the wave function of the universe, will be used. Moreover, the adiabatic treatment leads to an exact rewriting of the WDW equation in which the consequences of the dynamical character of gravity are displayed. It does not, in itself, introduce any kind of approximations nor does it require specific matter properties. Instead, it organizes the solutions of the WDW equation into series characterized by adiabatic parameters. Then, the validity of a truncation of these series requires that the parameters, say, be small. This in turn puts physical restrictions on the space of valid truncated solutions. The important fact is that this organization of the series answers by itself question 3, i.e. it both determines the precise conditions that guarantee that the classical and quantum descriptions coincide and explicitizes the mechanisms by which the two descriptions differ when the conditions are not met. In brief, the main condition that validates a truncation of the series is that the universe be macroscopic, i.e. that the matter sources driving gravity be macroscopiCnL5. (Recall that the appropriate character of the adiabatic treatment follows from the fact that one deals with dynamically coupled systems which are characterized by very different time-energy scales.) Then, the truncated adiabatic treatment shows that microscopic quantum transitions evolve according to a unitary evolution in the mean geometry parametrized by a, in a manner similar that non-adiabatic electronic (light) transitions occurring in molecules can be parametrized in terms of the (heavy) nuclei positions. Indeed, in cosmology, the rest mass of all matter delivers through the WDW constraint a kind of (macroscopic) inertia to gravity which fixes the geometry in which the (microscopic) transitions take place. Notice that the value of the Planck mass plays no role into this division in macroscopic and microscopic energy scales. We emphasize this point: for non empty macroscopic universes, it is unappropriate to develop the solutions of the WDW equation in series of the Planck mass, contrary to what is adopted in the "standard treatment" presented for instance in 5. Finally, the adiabatic treatment allows to question the problem of the interpretation of the solutions of the WDW equation in well defined mathematical terms. In the case of macroscopic universes, the coefficients that must be interpreted as the amplitudes to find the n-th state at a are designated by the formalism. As pointed out to me by S. Massar, this procedure to reach the interpretation of the WDW solutions bears many similarities to the original reasoning that Max Born used to reach the probabilistic interpretation of the solutions of the Schrodinger equatiod 6 .

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In both cases, it is through an examination of the mathematical properties of the solutions describing quantum transitions that the physical interpretation is, a posteriori, reached. The properties and the inferred interpretation are the following. 3, 14 • As long as matter is close to equilibrium and evolving in a macroscopic universe, the evolution extracted from the WDW equation coincides with the Schr6dingerian evolution of the corresponding problem. The identification of the amplitude to find matter in the n-th state is unambiguous and the physical interpretation follows from this identification. When matter is far from equilibrium but the universe still macroscopic, the evolution differs from the Schr6dingerian one, even though it is still unitary. As before, the identification and the interpretation of the amplitudes evolving unitarily are unambiguous. In both cases, unitarity follows from the ordered semi-classical expansion of the macroscopic universe. Therefore, there is a major difficulty when the cosmological expansion can no longer be described semi-classically. In this case indeed, through backscattering, one obtains non-vanishing coupling terms between quantum matter states associated with expanding and contracting universes. The present difficulties in interpreting the solutions of the WDW equation stem from the consequences of these coupling terms. Most probably, they require an extension of the usual concepts prevaling in quantum mechanics. We shall conclude this paper by presenting and commenting various approaches that have been proposed to overcome these difficulties. 2

The adiabatic treatment applied to quantum mechanics in classical cosmology

When studying quantum processes in classical cosmology, the expansion of the universe is treated at the background field approximation (BFA), i.e. a = a(t) is given from the outset and thus unmodified by the quantum transitions that one investigates. This is of course an approximation since, in general relativity, gravity is coupled to all forms of energy. To take into account this coupling is one of the jobs of quantum gravity. In classical cosmology, the point that is crucial for us is that the expansion law, a = a(t), leads, in the general case, to time dependent hamiltonians. Indeed only degenerate cases, like purely photonic homogeneous universes, are characterized by constants of motion. Therefore, in general, one deals with the (explicitly) time dependent Schr6dinger equation

i8t l1jJ(t))

= H(t)I1jJ(t))

(1)

(In this equation, as everywhere in this paper, we designate by t the proper time evaluated in the classical universe one is dealing with.) The main consequence of eq. (1) is that there will be no stationary states, a very useful feature on which we shall base our investigation of the role of time in cosmology. Physically it means that the expansion rate a induces quantum transitions. In this respect it should be noticed that the whole time dependence of H(t) comes from a(t) only, i.e.

H(t) = H(a(t))

(2)

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There is no external time dependence. In other words, we postulate that the universe is isolated. This possesses some flavor of General Relativity and will be automatically implemented when working in quantum cosmology. The adiabatic treatment consists in developing the solutions of eq. (1) in terms of instantaneous (normalized) eigenstates of H(a)

H(a)I'¢n(a»

= En(a)I'¢n(a))

('¢n(a)I'¢m(a)) = an,m

(3) (4)

One can already see the appropriateness of this treatment. It naturally incorporates the fact that the eigenstates and their eigenvalues depends on t through a(t) only. This is exactly like the red-shift mentioned in the introduction: the energy of photons measured in proper time scales as

W(a)

aemission = Wemission --a

(5)

which is a special case of eq. (3). By factorizing, as in time dependent perturbation theory 7 , the "free" kinetic factor exp( -i dt' En (t')) , the development in this basis reads

t

(6) n

By inserting this development into eq. (1), one obtains the equation that determines the time dependence of the coefficients Cn (t) :

OtCn =

L m#n

(Ot'¢m(t)I'¢n(t)) exp

[-i jt dt'(Em(t') - En(t'))] cm(t)

(7)

It is instructive to compare this equation with the one obtained in time dependent perturbation theory. In that case, the matrix elements (mlVln) of the perturbation V induce transitions among the "free" states In) and 1m). Here, these matrix elements are replaced by (Ot'¢m(t)I'¢n(t)). Simple algebra gives

('¢m IOtHI'¢n) En-Em

(8)

Thus, it is the time dependence of the instantaneous eigenstates, which is induced and determined by the time dependence of H, that induces, in turn, transitions among these states. Therefore, in classical cosmology, non-adiabatic transitions are caused by the expansion law a(t) since H(t) depends on time through a only. Moreover, one can rewrite eq. (7) directly in terms of a as

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One sees that the only place where time appears is in the difference of phases between neighboring states. Furthermore it appears parametrized by a, through the inverse rate l/a(a). Given this rate, the lapse of proper time from ao is of course given by a 1 (10) Llt(a) = da' -=----( ')

f

ao

a a

This equation makes explicit the fact that a( a) should be known to convert Doppler shifts controlled by areception/aemission into proper time lapses, c.f. the Introduction. As such eq. (9) is simply a rewriting of the Schrodinger equation in which the role of the expansion rate has been put forward. However, its real usefulness will become manifest in Section 3 since it is precisely in that guise that the evolution of the Cn, i.e. the weights of the instantaneous eigenstates, are delivered in quantum cosmology. This is not an accident, as we explain below. Moreover, eqs. (7, 9) prepare the analysis of the solutions both from the mathematical and the physical point of view. We start with the physical point of view. As noticed in the Introduction, Max Born was lead to the interpretation of the wave function by the properties of first order transition amplitudes. More precisely, the three following points are such that, according to him, "only one interpretation is possible,,16: 1. When initially Cm = 8(m, n), this means that the system is in the n-th state characterized by the eigenenergy En, 2. Mathematically, the final values of Cm are asymptotically constant and satisfy ~m Icm l2 = 1 (for all hermitian hamiltonians and for normalized eigenstates) and 3. Experimentally, the asymptotic "electron" is found in one of the outcomes labeled by n. These three properties are also found in the adiabatic treatment of I'if'(t)). This is not surprising in view of the analogies with perturbation theory. Thus, the properties of cn(t), solutions of eq. (7), can also be used to reach the interpretation of I'if'(t)) along Born's lines. Moreover, similar properties will be found as well in quantum cosmology when an extended adiabatic treatment is applied to the WDW equation. We shall then base our interpretation of the wave function of the universe on the properties so obtained. l,From a mathematical point of view, the first useful property of the adiabatic treatment follows from the fact that the instantaneous eigenstates form a "comoving" vector basis in Fock space. By comoving, we mean that no transition among eigenstates occurs in the limit in which the characteristic time of the expansion (i.e. a/a) is much larger than the characteristic time of quantum transitions (i.e. the time for the Golden Rule to be valid). To work in the adiabatic approximation simply means that one neglects completely these transitions. Moreover, to first order in the non-adiabaticity, the probability to find a transition takes a universal form controlled by an exponentially small factor, like in a tunneling process, see 14. The second useful property concerns the possibility of enlarging the dynamics. Up to now indeed, the adiabaticity concerned only the quantum dynamics of the (light) matter degrees of freedom since a(t) was treated at the BFA. However, the adiabatic treatment is naturally enlarged so as to take into account the quantum dynamics of the (heavy) degree a as well. This "extendible" property of the adiabatic treatment is precisely what we need to investigate (light) transitions in quantum

82

cosmology. Moreover, this treatment leads to a rewriting of the wnw equation such that the comparison to the Schrodinger equation is greatly facilitized. Indeed, the description of light transitions in quantum cosmology coincides with the BFA description when a first order (light) change is applied to the heavy WKB dynamics. To establish how these mathematical properties are precisely implemented in the formalism is the first aim of next Section. 3

The extended adiabatic treatment applied to quantum cosmology

As we explained, we shall use an extended adiabatic treatment as a guide to first identify and then to interpret the coefficient Cn(a), i.e. the weight of the n-th adiabatic state, that replaces the coefficient cn(a) of eq. (9). Before accomplishing this program, we briefly present the kinematical properties at work in quantum cosmology when the eigenstates are stationary, i.e. in the absence of transitions. It should be emphasized that this kinematical analysis is preparatory in character since it reveals the framework in which transitions take place upon considering non-degenerate cases. In those cases only, one can obtain a meaningful notion of evolution based on physical processes. 9.1

The (preparatory) notion of propagation in absence of transitions

To work with matter systems such that no transition occurs, requires that the matter states be stationary eigenstates of the hamiltonian H (a) H(a)l1f;n) aa l1f;n)

=

En(a)l1f;n)

(11)

0

In a Schrodinger context, this degenerate case would lead to no evolution in the sense that the coefficients en would be constant, see eq. (9). One simply obtains a (frozen) superposition of eigenstates whose relative phases depend on time through their eigenvalues. To give a physical substance to these phases requires either the addition of internal interactions or a coupling to the external world, see 13. In general relativity restricted to minisuperspace, when matter is characterized by an energy En(a), the gravitational action satisfies the Hamilton-Jacobi constraint 2

Ha () a

- -G (aa S a(a»2 + IW 2Ga

+ E n (a) -

2

+ Aa4

+

E ( )- 0 n a -

(12)

where G is Newton's constant, K, is equal to ±1 or 0 for respectively open, closed and flat three surfaces and A is the cosmological constant. The solution of this equation is simply Sn(a) = da'Pn(a') where the momentum of a driven by En(a) is (13)

r

The sign - (+) corresponds respectively to an expanding (contracting) universe. Upon working in quantum cosmology, the Hamilton-Jacobi constraint becomes the wnw equation (14) [Ha(a) + H(a)113(a» = 0

83

The matter states are still the stationary eigenstates of H(a) given in eq. (11). Therefore, as in Schrodinger case, the wave function Sea, ¢) can be decomposed as Sea, ¢) = (¢IS(a)) =

L Cn \lI(a; n) (¢I1/Jn)

(15)

n

where the weights Cn are constant and where the gravitational waves entangled to their corresponding matter state are solutions of (16) Being second order in 8a , each equation has two independent solutions. This has to be the case since classically we can work either with expanding or contracting universes. Indeed one verifies that the WKB waves (17) with positive (negative) Wronskians Wn

= \lI*(a; n) tt \lI(a; n)

(18)

correspond to expanding (contracting) universes in this semiclassical limit. It is now through the sign of the Wronskian rather than at the classical level that one can still choose to work either with expanding or with contracting universes (at least far from a turning point). Thus, upon abandoning the WKB approximation, one must deal with superpositions of contracting and expanding universes. As we shall see later in this article, this mixing leads to major difficulties concerning the notion of evolution. Before examining transitions, two aspects should be analyzed. First, we shall construct the Feynman kernel to go from ao, ¢o to a, ¢ in order to make contact with the notion of free evolution and with the classical theory. Secondly, we shall express the exact solutions of eq. (16) in terms of the WKB solutions eq. (17). Both aspects shall be exploited upon studying non-adiabatic transitions in quantum cosmology. The kernel to go from ao, ¢o to a, ¢ can be decomposed, as usual, with the help of the quantum conserved number n according to (19) n

The matter kernel Kn(¢; ¢o) is equal to (¢I1/Jn)(1/Jnl¢o) as in Schrodinger settings. The gravitational kernel Kn(a; ao) is a solution of eq. (12) and satisfies specific boundary conditions, see 7. In the WKB approximation, for a > ao, it is equal to (20)

At this point, we wish to point out that we used a dissymmetrical treatment of a and ¢ when constructing the kernel K or expressing the general solution of the WDW equation in eq. (15). As in Schrodinger settings, only the states of the matter field

84

0) comes from energy repartitions located near the saddle point of its phase. The location of this saddle point is given by the solution of

I:o

=

0

=

0

(23)

In the second equality, we have used the dispersion relation for the matter field to rewrite GE7r(4), E)IE=En by 1/¢n exactly as we just did with a. Eq. (23) means that the saddle value Eft is such that the lapses of time evaluated separately for gravity and matter agree. This constructive interference condition (see Box 25.3 in 12) can be conceived as the "dual" of a resonance condition in the following sense. In traditional time dependent settings, the dominant contribution to quantum processes arises from states such that the energy is conserved, c.f. the Golden RulE! 7 . Here, in quantum cosmology, energy conservation is built in, thanks to the constraint equation. Thus, the phases of sub-systems interfere constructively such that their classical times agree. This is exactly like the zeroth law of thermodynamics: at equilibrium, (inverse) temperatures agree. In physical terms, eq. (23) means that the cosmological time 6.t ft (a, ao) extracted from the expansion law (which has a similar status to that of the "ephemeris" time based on the solar system dynamics) is equal to the "cesium" time obtained from the (microscopic) behavior of matter. (This equality is not a tautology since it relates uncoupled dynamical systems characterized by widely separated time scales.) Notice finally that this condition of equal times can be formulated in purely classical terms. Indeed it provides the answer to the following questiorr1: Given ao, 4>0, what is the value of En such that 4> is reached at a ? Thus we have established that the emergence of time in the quantum kernel K, see eq. (21), makes use of classical concepts only: As in eq. (23), it follows, through a first order variation of E, from the classical relationship oEP(a, E) = 1/0,. The second point that we wish to address concerns the relationship between the exact solutions of eq. (16) and their WKB approximate expressions, eq. (17). This relationship is needed to properly investigate the consequences of quantizing the propagation of a expressed byeq. (14). Had we obtained a first order equation in Ga , this would have meant that a had no dynamics at all, like the longitudinal part of the electric field in classical electrodynamics (or in QED) which is fully determined by the charge density (operator). Being second order, eq. (14) implies that some backscattering might, and in general will, be spontaneously generated. This quantum effect cannot be expressed in terms of matter states unlike the CoulombCoulomb interaction which can be represented by a composite operator of charged fields. Moreover, gravitational backscattering will modify the propagation of matter

86

states as we shall see below. To express the exact solutions of eq. (16) in terms of their WKB expressions, we follow the usual technique of replacing a second order equation by a set of two coupled first order ones, see 14 for more details. The exact solution is first decomposed as

(24) and the the coefficients Cn(a) and Vn(a) are fully determined by requiring that oa "IJI(aj n) be instantaneously decomposable into purely forward and backward waves

iOa"IJI(aj n)

= Pn(a) [Cn(a)"lJlwKB(aj n) -

Vn(a)"IJIWKB(aj n)]

(25)

This guarantees that Cn(a) and Vn(a) are constant in the adiabatic limit OaPn/P~ ~ O. In addition, the Wronskian takes the simple form

Wn

= "IJI*(ajn)

t&a"IJI(ajn)

= ICn(aW

-IVn(aW

= constant

(26)

Simple algebra then yields the coupled first order equations

r da'Pn(a') V n(a) r da'Pn(a') Cn(a)

! OaPn(a)

-2i

! OaPn(a)

2i

2 Pn(a) e 2 Pn(a) e

(27)

These equations are equivalent to the original equation for "IJI(aj n), eq. (16). They constitute a convenient starting point for evaluating perturbatively non-adiabatic transitions from Cn(a) to Vn(a) (Le. backscattering). Moreover they resemble to the Schrodinger equation (9) that governs non-adiabatic transition in the particular case of two eigenstates with (oa"pm(a)l"pn(a)) replaced by OaPn(a)/Pn(a), see 14 for a more detailed comparison. 3.2

The double adiabatic treatment

In this subsection, we consider the non-degenerate cases in which the eigenstates of the matter hamiltonian depend on a. In these cases, the coefficients Cn also depend on a, like the cn(a) in eq. (9). To obtain the equation which governs their evolution, we need to join the usual adiabatic treatment presented in eqs. (3-9) with the treatment by wich eq. (16) is represented by eqs. (27). To this end, we first carry out the instantaneous diagonalization of H M , exactly like in eq. (3). We emphasize that this diagonalization does not require the "existence" of a Schrodinger equation. Using these instantaneous eigenstates, 13(a)), solution of eq. (14), can always be decomposed as

(28) n

The novelty of this decomposition with respect to eq. (15) is that the waves ..j has the same transformations under rotations and gauge transformations as a single RI' with (56)

Axiom 3 then allows one to conclude that (CK>" and eK>.. are structure constants)

[RK' R>..j

=L I'

(CK>..I' RI'

+ e K>..1' TI') + al'8q 0

[8/ (DQ

+ EA) + 8/ F1m]

(57)

109

where j, m, q E J.L. Although closure was postulated with respect to the old ks, we use the new ks here. This causes no difficulty because the two differ only by a superposition of T's, and such terms have been added anyway. When one operates with Eq. 57 on Ivac) one gets (58)

J.)

s:ands for a one-black hole state, a superposition of states with various J.L t. where Were R,.sR)..t Ivac) purely a two-black hole state as suggested by the field-theoretic analogy, one could not get Eq. 58. Inevitably (59)

with I •• ) a two-black hole state, symmetric under exchange of the "'8 and At pairs. The superposition of one and two-black hole states means that the rule of additivity of eigenvalues, Eq. 56, applies to one black hole as well as two: the sum of two eigenvalues of Q, }z or A of a single black hole is also a possible eigenvalue of a single black hole. For charge or z-spin component this rule is consistent with experience with quantum systems whose charges are always integer multiples of the fundamental charge (which might be a third of the electron's), and whose z- spins are integer or half integer multiples of n. This agreement serves as a partial check of our line of reasoning. In accordance with Axiom 1, let al be the smallest nonvanishing eigenvalue of A. Then Eq. 56 says that any positive integral multiple nal (which can be obtained by repeatedly adding al to itself) is also an eigenvalue. This spectrum of A agrees with that found in Sec. 5 by heuristic arguments. But the question is, are there any other area eigenvalues in between the integral ones (this has a bearing on the question of whether splitting of the levels found in Sec. 5 is at all possible) ? To answer this query, I write down the hermitian conjugate of Eq. 52: , 't _ 't [A, R,.] - -a,.R,.

(60)

Then

Thus differences of area eigenvalues appear as eigenvalues in their own right. Since A has no negative eigevalues, if n).. ::; n,., the operator Rl must anhilate the one-black hole state R)..lvac) and there is no black hole state RlR)..lvac). By contrast, if nIt < n).., Rl obviously lowers the area eigenvalue of R)... There is thus no doubt that RlR)..lvac) is a purely one-black hole state (a "lowering" operator cannot create an extra black hole: Eq. 61 shows that Rl anhilates the vacuum). In conclusion, positive differences of one-black hole area eigenvalues are also allowed area eigenvalues for one black hole. If there were fractional eigenvalues of A, one could, by substracting a suitable integral eigenvalue, get a positive eigenvalue below al, in contradiction with al's definition as lowest positive area eigenvalue. Thus the set {nal; n = 1,2,···} comprises the totality of A eigenvalues for one black hole, in complete agreement

110

with the heuristic arguments of Sec. 5 (but the algebra by itself cannot set the area scale ad. What about the degeneracy of area eigenvalues? According to Axiom 1, g(n), the degeneracy of the area eigenvalue nal, is independent of j, m and q. Thus for fixed {n""j"" m"" q",} where not all of j"" m", and q", vanish, there are g(n",) independent one-black hole states H",slvac) distinguished by the values of s. Analogously, the set {n>. = 1,j>. = O,m>. = O,q>. = O} specifies g(l) independent states H>.tlvac), all different from the previous ones because not all quantum numbers agree. One can thus form g(l)· g(n",) one-black hole states, [H", s, H>. tll vac), with area eigenvalues (n", + 1)al and charge and angular momentum just like the states H", slvac). If these new states are independent, their number cannot exceed the total number of states with area (n", + 1)al' namely g( n", + 1) ~ g(l)· g( n",). Iterating this inequality starting from n", = 1 one gets g(n} ~ g(l}n (62) The value g(l} = 1 is excluded because one knows that there is some degeneracy. Thus the result here is consistent with the law 29 which we obtained heuristically. In particular, it supports the idea that the degeneracy grows exponentially with area. The specific value g(l) = 2 used in Sec. 5 requires further input. Acknowledgments

I thank Slava Mukhanov for inspiring conversations and Avraham Mayo for discussions. This research is supported by a grant from the Israel Science Foundation established by the Israel National Academy of Sciences. References

1. J.D. Bekenstein, Lett. Nuovo Cimento 11, 467 (1974). 2. V. Mukhanov, JETP Letters 44,63 (1986); V Mukhanov in Complexity, Entropy and the Physics of Information: SFI Studies in the Sciences of Complexity, Vol. III, ed. W H Zurek (Addison-Wesley, New York, 1990). 3. J D Bekenstein in XVII Brazilian National Meeting on Particles and Fields, eds. A J da Silva, et. al (Brazilian Physical Society, 1996). 4. R. Ruffini and J.A. Wheeler, Physics Today 24,30 (1971). 5. P. Mazur, Gen. Rei. Grav. 19,1173 (1987). 6. D. Christodoulou, Phys. Rev. Lett. 25, 1596 (1970); D. Christodoulou and R. Ruffini, Phys. Rev. D 4, 3552 (1971). 7. For example see M Born, Atomic Physics (Blackie, London, 1969), eighth edition. 8. J D Jackson Classical Electrodynamics (Wiley, New York, 1962). 9. E M Lifshitz, L P Pitaevskii and V I Berestetskii, Quantum Electrodynamics (Pergamon, Oxford, 1982). 10. For further evidence see J D Bekenstein in The Black Hole Trail, eds. B. Bhawal and B. Iyer (Kluwer, Dordrecht 1998). 11. A.A. Starobinskii, Sov. Phys. JETP 37, 28 (1973). 12. S.W. Hawking, Phys. Rev. Lett. 26, 1344 (1971).

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13. J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973). 14. See the description of "Bekenstein's telephone number" for horizon elements in J A Wheeler, A Journey into Gravitation and Spacetime (Freeman, New York, 1990). 15. R.D. Sorkin, in Black Holes and Relativistic Stars, ed. R M Wald (University of Chicago Press, Chicago, 1998). 16. J.D. Bekenstein and V.F. Mukhanov, Phys. Lett. B 360, 7 (1995). 17. J.D. Bekenstein and V.F. Mukhanov in Sixth Moscow Quantum Gravity Seminar, eds. V A Berezin, V A Rubakov and D V Semikoz (World Publishing, Singapore, 1997). 18. U.H. Danielsson and M. Schiffer, Phys. Rev. D 48, 4779 (1993) 19. Ya.I. Kogan, JETP Letters 44, 267 (1986)]; I. I. Kogan, preprint hepth/9412232. 20. M. Maggiore, Nucl. Phys. B 429, 205 (1994). 21. C.O. Lousto, Phys. Rev. D 51, 1733 (1995). 22. M. Schiffer, "Black hole spectroscopy", Sao Paulo preprint IFT/P-38/89 (1989). 23. Y. Peleg, Phys. Lett. B 356, 462 (1995). 24. V. Berezin, Phys. Rev. D 55, 2139 (1997). 25. A.D. Dolgov and LB. Khriplovich, preprint hep-th/9703042. 26. J. Louko and J. Makela, Phys. Rev. D 54,4982 (1996). 27. A. Barvinskii and G. Kunstatter, Phys. Lett. B 289, 231 (1996). 28. J. Makela, preprint gr-qc/9602008. 29. H. Kastrup, Phys. Lett. B 385, 75 (1996). 30. J. Makela and P. Repo, preprint gr-qc/9708029. 31. M. Barreira, M. Carfora and C. Rovelli, Gen. Rei. Grav. 28, 1293 (1996). 32. K.V. Krasnov, Phys. Rev. D 55, 3505 (1997). 33. J. Makela, Phys. Lett. B 390, 115 (1997). 34. D. Page, Phys. Rev. D 13, 198 (1976). 35. W. Pauli, Z. Phys. 36,336 (1926). 36. E. Merzbacher Quantum Mechanics (Wiley, New York, 1970).

112

CURVATURE-BASED HYPERBOLIC SYSTEMS FOR GENERAL RELATIVITY YVONNE CHOQUET-BRUHAT Gravitation et Cosmologie Relativiste, t.22-12, Un. Paris VI, Paris 75252 France JAMES W. YORK, JR. and ARLEN ANDERSON Department of Physics and Astronomy, Univ. North Carolina, Chapel Hill NC 27599-9255 USA We review curvature-based hyperbolic forms of the evolution part of the Cauchy problem of General Relativity that we have obtained recently. We emphasize first order symmetrizable hyperbolic systems possessing only physical characteristics.

1

Introduction

Application of general relativity to fully three dimensional problems in astrophysics and cosmology has provided a driving force motivating further study and reformulation of Einstein's equations in 3+1 form. The vital role of the four constraint equations in setting up initial data for numerical evolution and in carrying out mathematical studies of gravitational field configurations has long been recognized. It led in the 1970's to a general, mathematically rigorous, and useful formulation of the constraints. (See, for example, the review.1 ) The standard evolution equations for the spatial metric g and the extrinsic curvature K were already known from a straightforward decomposition of the spacetime Ricci tensor (cf. the cases of zero shift; arbitrary shift~ and synthesis from the spacetime viewpoint and further developments 4 ). It was widely believed that these equations (first order in time derivatives; second order in space derivatives) were adequate for applications. However, they do not constitute a hyperbolic system leading to a proof of causal evolution in local Sobolev spaces of g and K into Einsteinian spacetime. Hence, further study of evolution equations is essential. In this paper, we review methods that we have used recently to obtain hyperbolic systems for the evolution of (g, K) with only physical characteristics and which propagate curvature along the physical light cone. We emphasize first order symmetrizable hyperbolic systems. 2

Einstein-Ricci Hyperbolic System

What we may call the "Einstein-Ricci" (ER) system is a spatially covariant hyperbolic formulation of the Einstein equations constructed from first derivatives of the spacetime Ricci tensor. The ER equations are obtained from the 3+1 form of the metric (1)

where N > 0 is the lapse, j3i is the shift, and the properly Riemannian metric gij is the spatial metric of a spacelike slice~. (~is understood to be a generic spacelike leaf of a foliation of a globally hyperbolic spacetime: locally, t=const.) It is convenient to work in the cobasis (}o = dt, (}i = dx i + j3i dt, with d(}o; = -~co;/3'Y(}/3 A(}'Y and only CiOj = -CijO = 8j (3i # O. The connection one-forms wo; /3 = 'Yo; /3'Y(}'Y are given by the connection coefficients 0;

_

'Y /3"1 -

ro;

/3"1

+ 9 0;6C' 6(/3g'Y)' - 2"1 Co; /3"1'

(2)

113

where r denotes a Christoffel symbol. In particular, the extrinsic curvature of ~ is Kij == - N 1'0 ij, or, upon evaluation of1'°ij, (3) where 8o = 8 t - £{3 evolves t-dependent spatial objects in the direction orthogonal to ~, and £{3 is the Lie derivative in ~ along the shift vector. The relation (3) serves as the evolution equation for gij. Letting an overbar signify spatial objects, we note for future reference that gij = gij, gij = gi j , and 1'ijk = r i jk = f'i jk . For completeness, we also state 1'ioo = N'IliN, 1'°00 = N- 18 0N, 1'°Oi = 1'°iO = N- 1'IliN, 1' k oi = -N Kk i, and 1'k iO = 1'k Oi + 8i (3k. The 3+1 decomposition of the spacetime Riemann tensor is given, for example, in Ref. [4]. Likewise, the Ricci tensor is given identically by Rij - N- 18 0K ij

+ HKij

N'Ilj(Hg ij - K ij ), N80H - N 2KmkK mk

+ N'Ilk'llk N ,

where H == Kk k == K. The heart of the ER system 5fl ij

== 8 0 R ij

(4) (5)

- 2KikKkj - N-1'lli'lljN,

7

(6)

is defined by (7)

- 2'1l(i R j)0.

Upon working out this identity using (4)-(6) and substituting the Einstein equations

R",{3

=

P",{3 = 87r(T",{3 - ~g",{3T"Y"y), with G = c = 1, we obtain the equation

(8) where the physical wave operator for arbitrary shift is tJ == -N- 18 0 N- 1 80 + 'Ilk'll k. We study the vacuum case fl ij = here. In general, the value of fl ij is defined by matter fields. The detailed form of the right hand side of (8) can be found in Refs. [5]-[7]; the present conventions are those in Refs. [6],[7]. Here we point out that J ij = Jij[2 in g; 2 in N;l in K]. (The numbers in the bracket indicate the highest order derivatives that occur.) The slicing-dependent term 8ij is given by

°

(9) We observe that the term 'Ili'lljH = gmk'lli'lljKmk would spoil the hyperbolicity of (8); therefore, 8 ij must be set equal to a functional involving fewer than two derivatives of K ij . Notice that (10) where 9 = detgij > 0, 1'0 == (4)g",{31'0",{3, and a == g-lj2N is a lapse function of weight 1, first introduced in the "algebraic gauge"S and subsequently as a "proper-time gauge,,9 and in connection with the "new variables" program for general relativity in Hamiltonian form,lO where a = !:f is called the "densitized lapse." The algebraic gauge has a simple relationship to "harmonic time slicing," 1'0 = 0, as we see in (10). One most easily deals with 8 ij by freely specifying a(t, x) > 0. (The shift (3i(t, x) is also arbitrary.) A "gauge source" f(t, x) can be used, as observed by Friedrich (see Ref. [11] for references); here, f(t,x) = 80 loga. Then the first term in (10) yields a general harmonic time slicing equation for N, namely (11)

114

or equivalently,

(12) In this case, the system composed of (12) and the third-order equation resulting from combining (3) and (8), whose principal term is 080 gij , is quasi-diagonal hyperbolic 12 for the unknowns g and g-I/2 N. Therefore, the "third-order" ER equations have a unique, well-posed solution in a suitable Sobolev function space (local in time and in space; global in space can be obtained as well), given appropriate initial data~,13 We wish to verify that every solution of the third order ER system, equivalently ((3), (8), (12) or (II)}, is a solution of the Einstein equations given suitable initial data. Thus, suppose g, K, and N satisfy these equations. Then, by using the contracted Bianchi identities \1(3G(3o. == 0, and these ER equations, it follows that G(3o. = if g and K satisfy initially the usual momentum and Hamiltonian constraints (C i == -2N- 1 Roi = 0, C = 2Goo = 0), if BoKij satisfies Rij = initially, and if initial data {a > O,)3i} or {N > O,)3i} are given~,6 Conversely, because every globally hyperbolic metric solution of Einstein's equations can be given in a harmonic time slicing,1 it follows that all globally hyperbolic solutions of Einstein's equations can be reached by solving this Einstein-Ricci system. The ER system as given holds for any spacetime dimension. For four dimensions, it can be written in first-order symmetrizable hyperbolic (FOSH) form. To put a system in first-order form, one introduces auxiliary variables in place of derivatives of the fundamental variables and adds additional equations to describe the evolution of these new variables. The crux is whether this process terminates. Generally existence of a higher order wave equation on the fundamental variables is necessary to halt the process. Though we have a wave equation for K, presence of second spatial derivatives of g and N in Jij prevent the reduction to first order form from being obvious. However, the second derivatives of g occur only in Rij and Rijkl terms; the latter, in three space dimensions, can be reduced to R ij , which in turn can be eliminated from Jij by using (4). First derivatives of g are handled by an evolution equation5,7 for the Christoffel symbol :r (inadvertently omitted in [6]). On the other hand, to handle the second derivatives of N, one first finds a wave equation for N. Then, by applying Vi to it, a wave equation for ViN, or ai = VilogN, is obtained. From this point, reduction to first order form is straightforward.5-7. [The wave equation for N follows from applying Bo to (11) to get an equation with BoBoN and BoH. Using (6), we then find a nonlinear wave equation that can be written in terms of LJN. Then ViLJN gives the wave equation LJai. See Refs. [5), [7) for details.) One then shows that the first order system is symmetrizable and hyperbolic. In this system, the spacetime metric (N, gij) evolves at zero speed along the direction 80 orthogonal to t=constant. Quantities with dimensions of curvature propagate at speed 1 ("e") along the physical light cone. There are no unphysical speeds or directions as there are in many of the FOSH systems that have so far appeared in the literature.l l ,14,15 Of course, we except the FOSH form of the ER system5- 7 (described here) and the mathematically equivalent, but more transparent, "Einstein-Bianchi,,16,17 (EB) system presented in the next section. Having only physical characteristics is essential for an ideal match of physics and mathematics. In practice, it is also useful: for example, consider obtaining the gravitational radiation content of a numerical Cauchy evolution on a finite grid. The radiation is propagating at light speed and is either "extracted" 18 or "evolved to null infinity" 19 based on field values at a finite distance from the source. We know that gravitational radiation is curvature propagating at light speed and that

°

°

115 is what the ER system describes. The description is even more explicit and transparent in the EB system. A "fourth order" ER system also exists~O One forms the expression

(13) to obtain it. It has a well-posed Cauchy problem and is hyperbolic ("non-strict,,21) for any N > 0 and f3 i . It has been applied to the non-linear perturbative regime of high frequency wave propagation 22 ,23 and has been used to obtain an elegant derivation of gauge-invariant perturbation theory for the Schwarzschild metric~4 3

Einstein-Bianchi Hyperbolic System

The "Einstein-Bianchi" (EB) system discussed next was given first in [16] and with energy estimates and full mathematical detail in [17]. It is similar to an analogous system, obtained by H. Friedrich,ll that is based on the Weyl tensor and is causal but with additional unphysical characteristics. Recall that the Riemann tensor satisfies the Bianchi identities

(14) where we use a comma to stress the two separate antisymmetric index pairs (not to indicate partial differentiation). These identities imply by contraction and use of the symmetries of the Riemann tensor (15) V' ",R'" {3,),,/J. + V' /J.R{3)" - V' )"R{3/J. == 0, where the Ricci tensor is defined by R"'{3,"'/J.=R{3/J..

If the Ricci tensor satisfies the Einstein equations

(16) then we have V' ",R'" {3,),,/J. = V' )"P{3/J.

-

V' /J.P{3)".

(17)

Equation (14) with {af3'Y} = {ijk} and (17) with f3 = 0 contain only derivatives of the Riemann tensor tangent to ~; hence, we consider these equations as constraints ("Bianchi constraints"). We shall consider the remaining equations in (14) and (17) as applying to a double two-form A",{3,),,/J.' which is simply a spacetime tensor antisymmetric in its first and last pairs of indices. We do not suppose a priori a symmetry between the two pairs of antisymmetric indices. These "Bianchi equations" are

(18) V'OAOi,),,/J. + V'hAhi,),,/J. = V')"Pi/J. - V' /J.Pi)" == Ji,),,/J., (19) where the pair [AJ.!] is either [OJ] or [ill. We next introduce following Bel25 two "electric" and two "magnetic" space tensors associated with the double two-form A, in analogy to the electric and magnetic vectors associated with the electromagnetic two-form F. That is, we define the "electric" tensors by (20)

116 _ 1 hk,lm = 41]ihk1]jlmA ,

Dij

while the "magnetic" tensors are given by Hij Bji

_ 1 -1 hk = 2N 1]ihk A ,OJ,

==

1

1

2 N - 1]ihkAOj,

hk

(21)

.

In these formulae, 1]ijk is the volume form of the space metric g. We note that: (1) If the double two-form A is symmetric with respect to its two pairs of antisymmetric indices, then Eij = E ji , Dij = D ji , and Hij = B ji . (2) If A is a symmetric double two-form such that A",6 == AA",A,6 = cg",6, then Hij = Hji = Bji = Bij and Eij = D ij . Property (1) is obvious and (2) follows from the "Lanczos identity.,,26 (We note that 1]Oijk = N1]ijk relates the spacetime and space volume forms in using the Lanczos identity.) In order to extend the treatment to the non-vacuum case and to avoid introducing unphysical characteristics in the solution of the Bianchi equations, we will keep as independent unknowns the four tensors E, D, B, and H, which will not be regarded necessarily as symmetric. The symmetries will be imposed eventually on the initial data and shown to be conserved by evolution. We now express the Bianchi equations in terms of the time dependent space tensors E, H, D, and B. We also express spacetime covariant derivatives \7 of the spacetime tensor A in terms of space covariant derivatives ~ and time derivatives 80 of E, H, D, and B by using the connection coefficients in 3+1 form as given earlier. The first Bianchi equation (18) with [AJ.ll = [Ojl has the form (22)

A calculation yields (22) in the form 80(1]ihkHij)

(Ldhk,j

-

+ 2N~[hEklj + (L 1 )hk,j

1

NK j 1]\k H i/

= 0,

+ 2(~[hN)Eklj + 2N1]iljKI[kBhli

(23) (24)

_(~I N)1]ihk1]mljDim.

The second Bianchi equation (19), with [AJ.ll = [OJ], has the form (25) where J is zero in vacuum. A calculation similar to the one above yields for the second Bianchi equation (26) -N(TrK)Eij +NKkjEik

-(~hN)1]hliHlj

+ 2NKi k E kj

+N

Kkh1]I\1]mkjDlm

(27)

+ (~k N)1] l kj B il .

The non-principal terms in the first two Bianchi equations (23) and (26) are linear in E, D, B, and H, as well as in the other geometrical elements NK and ~ N. The characteristic matrix of

117 the principal terms is symmetrizable. The unknowns Ei(j) and Hi(j) , with fixed j and i = 1,2,3 appear only in the equations with given j. The other unknowns appear in non-principal terms. The characteristic matrix is composed of three blocks around the diagonal, each corresponding to one given j. The lh block of the characteristic matrix in an orthonormal frame for the space metric g, with unknowns listed horizontally and equations listed vertically, (j is suppressed) is given by

EI (26h (26h (26)3 (23h3 (23hl (23h2

~o

E3 0 0

0 -N6 0 N6

N6 -N6 0

E2 0

~o

0 0 0 N6 -N6

[

~o

HI 0 -N6 N6 ~o

0 0

H2 N6 0 -N6 0 ~o

H3 -N6 N6 0 0 0

0

~o

)

(28)

This matrix is symmetric and its determinant is the characteristic polynomial of the E, H system. It is given by _N6(~0~0)(~a~a)2 . (29) The characteristic matrix is symmetric in an orthonormal space frame and the timelike direction defined by 80 has a coefficient matrix To that is positive definite (here To is the unit matrix). Therefore, the first order system is symmetrizable hyperbolic. We will not compute the symmetrized form explicitly here. The second pair of Bianchi equations is obtained from (18) and (19) with [>'J.t] = [lm]. We obtain from (18) 80(r/hkrflmDii) (L 3)hk,lm

+ 2Nrflm ti'[kBh]i + (L 3)hk,lm

= 0,

== 2Nrti[mKil]",ihkDin + 2",i1m(ti'[kN)Bh]j +2NKI[hEk]m

(30) (31)

+ 2NKm[kEh]1 + 2Hi[I(ti'm]N)"'\k'

Analogously, from (19) we obtain

(32) -N(TrK)",i1mBij + 2N",hi[mKil]Bih + 2NK\rflmBhi _(ti'jN)",hi i",n1mDhn - 2N",i hiHj[mKhl] + 2Ei [m ti'l]N.

(33)

Consider the system (30) and (32) with [lm] fixed. Then j in "'ilm is also fixed. The characteristic matrix for the [lm] equations, with unknowns Dij and B ij , j fixed, with an orthonormal space frame, is the same as the matrix (28). If the spacetime metric is considered as given, as well as the sources, the Bianchi equations (23), (26), (30), and (32) form a linear symmetric hyperbolic system with domain of dependence determined by the light cone of the spacetime metric. The coefficients of the terms of order zero are ti' Nor NK. The system is homogeneous in vacuum (zero sources).

118

4

Determination of (f', K) from Knowledge of the Bianchi Fields

We next link the metric and connection to our Bianchi fields. This link uses and extends an idea introduced by Friedrich l l in his Weyl-tensor construction mentioned above. We will need the 3 + 1 decomposition of the Riemann tensor, which is4

Rij,kl Roi,jk

.R;j,kl + 2Ki[k K IJj' 2NV[jKkJi'

Roi,Oj

N(80Kij

(34) (35)

+ NKikKkj + ViojN).

(36)

From these formulae one obtains those for the Ricci curvature given in Sec. 2: (4), (5), and (6), where, in this section, we will not denote Kj j = TrK by H. The identity (3) and the expression for the spatial Christoffel symbols give

(37) Therefore, from the identity (35), we obtain the identity

8of'\j + NVhKij =

KijOhN - 2Kh(ionN - 2Ro(i,n h •

(38)

On the other hand, the identities (36) and (4) imply the identity

80 K ij + NRij + VjoiN == -2NROi,Oj -

N(TrK)Kij

+ NR ij .

(39)

We obtain equations relating f' and K to a double two-form A and matter sources by replacing, in the identities (38) and (39), Ro(i,n h by (AO(i,j)h + A\j,i)O)/2, ROi,oj by (AOi,oj + AO j ,Oi)/2, and the Ricci tensor of spacetime by a given tensor p, zero in vacuum. The terms involving A are then replaced by Bianchi fields E, H, D and B. The first set of identities (38) leads to equations with principal terms

(40) To deduce from the second identity (39) equations which will form together with the previous ones a symmetric hyperbolic system, we set, we use algebraic gaugeS, as in Sec. 2, by setting

(41) where a is a given positive scalar density of weight minus one, a function of (t, Xi). (Note that the present a is the a-I of Refs. [16] and [17].) The lapse N is now a derived quantity depending on gl/2 and on a. The use of a, if f' denotes the Christoffel symbols of g, implies that

f'\h = Oi log N - Oi log a.

(42)

The second set of identities (39) now yields the following equations, where N denotes gl/2a,

80 K ij + NOhf'h ij

=

+ Oi loga)(f'kjk + OJ log a)] N(Eij + E ji ) - N(TrK)Kij + N Pij.

N[f'm ihf'h jm - (f'\h

-N(OiOj log a - f'k ijOk log a) -

(43)

119 The first set (38) yields

80 f\j + NVh Kij

=

+ Ok log a) + oj) log a) - N(r/(/ Bi)k + Hk(i1J k j)h).

N Kijghk(fm mk

-2NKh(i(fmj)m

(44)

We see from the principal parts of (43) and (44) that the system obtained for K and f has a characteristic matrix composed of six blocks around the diagonal, each block a four-by-four matrix that is symmetrizable hyperbolic with characteristic polynomial N4(~0~0)(~a~a). The characteristic matrix in a spatial orthonormal frame has blocks of the form

N6 N6

o

~o

o o

5

(45)

~o

o

Symmetric Hyperbolic System for (E,H,D,B,g,K,f)

We denote by S the system composed of the equations (23), (26), (30), (32), (3), (43), and (44), where the lapse function N is replaced by gl/2a. This system is satisfied by solutions of the Einstein equations whose shift {3, hidden in the operator 00, has the given arbitrary values and whose lapse has the form N = gl/2 a . (Clearly, any N > 0 can be written in this form.) From the results of the previous sections, we see that for arbitrary a and {3, and given matter sources p, the system S is a first order symmetrizable hyperbolic system for the unknowns (E, H, D, B, g, K, f). Note that the various elements E, H, D, B, g, K, and f' are considered as independent. For example, a priori, we neither know that f denotes the Christoffel symbols of g nor that E, H, D, and B are identified with components of the Riemann tensor of spacetime. We now consider the vacuum case. The original Cauchy data for the Einstein equations are, with ¢ a properly Riemannian metric and 1/J a second rank tensor on an initial spacelike slice ~o,

The tensors ¢ and

glo = ¢, Klo = 1/J. 1/J must satisfy the constraints, which read in vacuum,

(46)

Roi =

(47)

0= Goo

0,

1

Roo - -gooR 2 Roo + ~ N 2 ga{3 Ra{3 1 2 .. 2N (R - KijK'J

+ (TrK)

(48)

2

),

with R = gij Rij. The initial data given by ¢ determine the Cauchy data fh ij 10 and thus Rij,k!lO. Then, R;j,k!lo is determined by using also 1/J. To determine the initial values of the other components of the unknowns of the system S, we use the arbitrarily given data a and {3. In particular, we use N = gl/2 a to find Roi,jklo and to compute VjoiNlo appearing in the

120

identity (4). We deduce from (4) 8oKijlo when Rij = 0, which enables Roi,Oj to be found from (36). All of the components of the Riemann tensor of spacetime are then known on ~o. We identify them with the corresponding components of the double two-form A on ~o: the latter have thus initially the same symmetries as the Riemann tensor. We find the initial values of (E, H, D, B) according to their definitions in terms of A. Detailed existence theorems and proofs that solutions of the EB system, with initial data given as in the preceding paragraph are found in [18]. Here we give a less detailed argument, which is nevertheless completely rigorous. Let the initial data for the vacuum EB system be given as above. We know that our symmetrizable hyperbolic system has a unique solution. Because a solution of Einstein's equations in algebraic gauge [i.e., with given aCt, x)]-previously proven to exist 5 -together with its connection and Riemann tensor, satisfies the present EB system while taking the same initial values, that solution must coincide with the solution of the present EB system in their common domain of existence. The FOSH form of the ER system and the EB system are completely equivalent mathematical systems, with the EB system perhaps having its unknowns arranged in a more transparent way. For instance, the characteristic Bianchi fields with respect to a fixed space direction that propagate along the light cone are the directionally transverse components of the Riemann tensor. For a detailed discussion of charateristic fields of the FOSH form of the ER system-also curvatures-see [7]. These fields for the EB system will be discussed in more detail elsewhere, as will "third order" and "fourth order" forms of the EB system, analogous to those of the ER system. 6

A Concluding Remark About Constraint Violations

It is well known that the standard 3+1 equations (3)-(6), with time derivatives 80g ij and 80Kij only, were reformulated in terms of closely related variables and written in Hamiltonian form in far-reaching work by Arnowitt, Deser, and Misner27 (ADM) and by Dirac?8 Indeed, the standard 3+1 equations are often referred to as the "ADM equations." However, it can be shown29 ,3o that when there are violations of the constraints (as there generically are in numerical work), the standard 3+1 evolution equations and the ADM evolution equations are not equivalent. From an analytic standpoint, the standard 3+ 1 equations in terms of (g, K, R;j, Roi, GO 0) produce a first order symmetric hyperbolic system for the constraint violations, to which suitable "energy" bounds for the growth of these violations can be applied, while the actual ADM equations in terms of (g, 7r, G1''') do not produce a hyperbolic system for evolution of constraint violations. However, neither the standard 3+1 equations nor the ADM equations are known to be well-posed, independently of the issue of constraint conservation. The ER and EB systems are well-posed, as are the equations describing conservation of the constraints in these systems. This subject is treated in more detail elsewhere.31

Acknowledgment J.W.Y. and A.A. were supported by National Science Foundation grants PHY-9413207 and PHY 93-18152/ASC 93-18152 (ARPA supplemented). 1. Y. Choquet-Bruhat and J.W. York, in Geneml Relativity and Gmvitation, I, edited by A.

Held (Plenum: New York, 1980), pp. 99-172.

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2. A. Lichnerowicz, Problemes Globaux en Mecanique Relativiste, (Hermann: Paris, 1939). 3. Y. Choquet (Foures)-Bruhat, J. Rat. Mechanics and Anal. 5,951-966 (1956). 4. J.W. York, in Sources of Gravitational Radiation, ed. L. Smarr, (Cambridge Univ. Press: Cambridge, 1979). 5. Y. Choquet-Bruhat and J. W. York, C. R. Acad. Sci. Paris, t. 321, Serie I, 1089 (1995). 6. A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.W. York, Phys. Rev. Lett. 15, 3377 (1995). 7. A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.W. York, Class. Quantum Grav. 14, A9-A22 (1997). 8. Y. Choquet-Bruhat and T. Ruggeri, Commun. Math. Phys. 89,269-275 (1983). 9. C. Teitelboim, Phys. Rev. D25, 3159-2176 (1982); Phys. Rev. D28, 297-309 (1983). 10. A. Ashtekar, New Perspectives in Canonical Gravity, (Bibliopolis: Naples, 1988). 11. H. Friedrich, Class. Quantum Grav. 13, 1451-1459 (1996). 12. A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.W. York, to appear in Proceedings of the 1996 Texas Symposium on Relativistic Astrophysics, gr-qc/970301O (1997). 13. J. Leray, Hyperbolic Differential Equations, lecture notes, Institute for Advanced Study, Princeton (1952). 14. A. Fischer and J. Marsden, Commun. Math. Phys. 28, 1-38 (1972). 15. S. Frittelli and O. Reula, Commun. Math. Phys. 166, 221-235 (1994); Phys. Rev. Lett. 16, 4667-4670 (1996). 16. Y. Choquet-Bruhat and J. W. York, Banach Center Publications, 41, Part 1, 119-131 (1997). 17. A. Anderson, Y. Choquet-Bruhat and J.W. York, to appear in Topological Methods in Nonlinear Analysis, gr-qc/9710041, (1997). 18. Binary Black Hole Grand Challenge Alliance: M. Rupright, A. Abrahams, L. Rezzolla, et al., to appear in Phys. Rev. Lett., gr-qc/9709082, (1997); M. Rupright, A. Abrahams, and L. Rezzolla, gr-qc/9802011 (1998). 19. Binary Black Hole Grand Challenge Alliance: R. Gomez, L. Lehner, R. Marsa, J. Winicour, et al., gr-qc/9801069 (1998). 20. A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.W. York, C.R. Acad. Sci. Paris, t. 323, Serie II b, 835-841 (1996). 21. J. Leray and Y. Ohya, Math. Ann. 110,167-205 (1967). 22. Y. Choquet-Bruhat, in Modem Group Analysis, eds. N. Ibragimov and F. Mahomed (World Scientific: Singapore), to appear (1997). 23. Y. Choquet-Bruhat and A. Greco, Circolo Matematico di Palermo, to appear (1997). 24. A. Anderson, A. Abrahams, and C. Lea, gr-qc/9801071 (1998). 25. L. Bel, C. R. Acad. Sci. Paris 246, 3105 (1958) and Thesis, University of Paris (1961). 26. C. Lanczos, Ann. of Math. 39, 842-850 (1938). 27. R. Arnowitt, S. Deser, and C.W. Misner, in Gravitation: An Introduction to Current Research, ed. L. Witten, (Wiley: New York, 1962), pp. 227-265. 28. P.A.M. Dirac, Proc. Roy. Soc. London A246, 333-343 (1958). 29. S. Frittelli, Phys. Rev. D55, 5992-5996 (1997). 30. A. Anderson and J.W. York, unpublished. 31. A. Anderson and J.W. York, "Propagation of constraint violations in Hamiltonian formulations of general relativity," in preparation.

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INNER STRUCTURE OF SPINNING BLACK HOLES AMOS OR! Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel We present the recent results concerning the structure of the singularity inside generically-perturbed spinning black holes. We first review the perturbative approach, and describe the main results obtained from it. Then, we briefly review two other approaches to the problem: (i) a non-perturbative local approach, and (ii) the numerical approach. These three approaches yield consistent results, and lead to a coherent picture of the singularity at the Cauchy horizon of generic spinning black holes. The singularity is found to be null, weak, scalar-curvature, and (in the presence of non axially-symmetric perturbations) oscillatory.

In this lecture we shall discuss the inner structure of spinning black holes (BHs), and, in particular, the structure of the spacetime singularity located inside such BHs. The consideration of spinning BHs has an obvious motivation, because the realistic astrophysical BHs in our universe are spinning 1 ,2 j and, as we shall see, the spin of the BH has a crucial effect on its inner structure. Since the late 1960s we know from the singularity theorems (by Penrose, Hawking and others) that there should be some sort of spacetime singularity inside black holes. But until a few years ago, the location and features of this singularity were unclear for realistic spinning black holes. In the last decade there has been a very significant progress in this area; and, as a consequence, it appears that by now we have a fairly reasonable understanding of the location, structure, and main features of the spacetime singularity that forms in the gravitational collapse of realistic spinning objects (at least in some portion of the singular hypersurface). The purpose of this talk is to review these new developments, and to describe the present understanding of the singularity inside realistic spinning black holes. We shall review here three approaches to the problem: The perturbative approach, the local analytic approach, and the numerical approach. So far, the perturbative approach yields the most decisive and most informative results concerning the structure of the singularity. Therefore, the main focus of this lecture will be on the perturbative approach (section 2) and what we learn from it about the features ofthe singularity (section 3). Later, we shall also describe, more briefly, the progress in the local analytic approach (section 4) and in the numerical approach (section 5). The results obtained from these two latter approaches are less decisive or informative, but are nevertheless fully consistent with the results of the perturbative approach. 1

Background

The simplest solution describing a spinning BH is the Kerr geometry. Fig. 1 describes the Penrose diagram of this spacetime. As a reference, we show in Fig. 2 the Penrose diagram of a Schwarzschild BH. One immediately notices the drastic difference between these two spacetimes. The green curves represent typical worldlines of infalling particles. In Schwarzschild, the infalling particle is doomed to crash on the

123 A

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"

V /

A

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Figure 1: Penrose diagram of a Kerr black hole.

central spacelike singularity. In Kerr, the singularity is timelike, and the particle's worldline does not intersect it: Instead, it is ejected (through a white hole) into another external universe - a phenomenon known as gravitational bounce 3 . This difference in the causal structures of the two spacetimes clearly indicates that the spin of the BH may have a drastic effect on its inner structure. One still needs to address the following question: To what extent is the inner structure of the pure Kerr geometry relevant to realistic spinning BHs? When considering this question, we must take into account two important considerations: 1. The Kerr geometry is stationary.

On the other hand, the spacetime produced in (non-spherical) gravitational collapse is in general dynamic, i.e. nonstationary. Therefore, it cannot be the exact Kerr geometry (rather, it may be viewed as K err+ perturbation, as we discuss below).

2. In the spacetime diagram of Kerr there is a Cauchy horizon (CH) - the blue null hypersurface in Fig. 1. Penrose 4 realized already in the late 1960s that infalling fields are infinitely blue-shifted at the CH, and consequently the energy-density of such fields diverges at the CH. When these two facts are combined, we realize that in a realistic BH the ingoing perturbations may convert the CH into a curvature singularity. The CH of Kerr is thus unstable in this sense. Our goal is to analyze the behavior at the CH, and to find out what is the outcome of this instability: Does a spacetime singularity really form there? And, if so, what type of singularity? Is it strictly null (like the unperturbed CH), or spacelike (like e.g. the Schwarzschild or BKL singularity)?

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Figure 2: Penrose diagram of a Schwarzschild black hole.

Answering the above questions is not a simple taskj The analysis is complicated because, first, the model is not spherically-symmetric, and second, the field equations, which are just the vacuum Einstein equations, are nonlinear. In order to simplify the analysis, in the last few decades people often considered a simplified toy-model of a spherical charged BH, instead of a spinning uncharged one. (This model is obviously simpler, due to its spherical symmetry.) This model has a strong motivation, as we can see from Fig. 3, which displays the Penrose diagram of the Reissner-Nordstrom (RN) geometry. In this geometry, too, the singularity is timelikej and the infalling particle avoids it, and is ejected into another external universe. Also, in the RN geometry infalling fields develop infinite blueshift at the CH, indicating the instability of the latter. One may therefore hope that understanding the instability of the CH in spherical charged BHs will help us understand the situation in spinning BHs as well. I will now briefly outline the important stages in the investigation of the instability of the CH inside spherical charged BHs during the last few decades: • Basic geometrical-optics considerations: Penrose (1967) showed that the CH of RN is a locus of infinite blue shift, suggesting that the CH is unstable 4 . • Linear stability analysis of the RN geometry, with respect to various massless fields (scalar, electromagnetic, and linearized gravitational fields): Simpson & Penrose (1973) 5, Giirsel al (1979) 6, Chandrasekhar & Hartle (1982) 7, and others. [See also the analysis of a linear scalar field in Kerr, by Novikov and Starobinsky (1980) 8.J Basically, all these analyses led to the same conclusion: The infalling linear fields develop a singularity at the CH, where these fields

et

125 A

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v /

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Figure 3: Penrose diagram of a Reissner-Nordstrom black hole.

(or their gradients) blow up. Consequently, the energy-momentum density of these linear fields diverges at the CH. • Simplified nonlinear toy models - spherical charged black holes perturbed by null fluids: Hiscock (1981) 9 considered an ingoing null fluid, and showed that a null curvature singularity forms at the CH. Later, Poisson & Israel (1989) 10 developed the mass inflation model, in which there are two null fluids - ingoing and outgoing. They showed that the mass function diverges at the CH, which indicates a scalar-curvature singularity (namely, the curvature scalar Riemann2 diverges). Then, Ori (1991) 11 analyzed a simplified version of the mass-inflation model, in which the outgoing null fluid is "compressed" into a thin null layer. This model recovered the previous results - a null singularity forms at CH, and the mass function diverges there. In addition, this model allowed the calculation of the asymptotic form of the metric functions near the CH. Quite surprisingly, it was found that the singularity is weak (in Tipler's terminology 12). Namely, the tidal deformation experienced by an infalling observer is finite (and typically also very small) as the observer hits the singularity. To summarize, the picture emerges from the above simplified, spherical, nonlinear toy models of perturbed charged BHs is the following: In the perturbed BH, the CH becomes a spacetime singularity which is null, weak, and scalar-cunJature (massinflation) . Equipped with this insight from spherical charged BHs, we now turn back to the more realistic problem - the inner structure of (uncharged) spinning BHs. To

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Figure 4: Schematic Penrose diagram describing the gravitational collapse of a generic spinning object.

that end, we shall employ the perturbative approach, as we describe in the next section. 2

Generic Spinning Black Holes - The Perturbative Approach

One of the major difficulties in analyzing the geometry inside a non-stationary BH is the strong nonlinearity of the relevant field equations, i.e. the Einstein equations. In order to overcome this difficulty, we can employ the perturbative approach, as we now explain. Consider the gravitational collapse of a generic (uncharged) spinning object (a "star"). The object may have a generic shape, and need not be axially-symmetric. Initially, the geometry will by time-dependent, as the star's various (gravitational) multipole moments will emit gravitational waves. We do expect, however, that after the completion of the collapse the geometry will settle down to Kerr asymptotically. This is the essence of Wheeler's principle that "black holes have no hair". Thus, at late time, the geometry outside the black hole may be viewed as Kerr+perturbations. This situation is schematically described in Fig. 4, in which the collapsing object is marked by brown, the dark gray marks areas with strong deviations from Kerr, and the bright regions outside the BH are regions of weak perturbations, i.e. weak deviations from Kerr. The perturbations become arbitrarily weak on the approach to the point P, which represents the timelike infinity of the external universe. This region of arbitrarily weak perturbations (the "neighborhood" of P) is denoted in Fig. 4 by the blue circular curve, and we shall refer to it as the late-time region. For a (nearly) spherical BH, the perturbations outside the BH were found to

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v=O CH

L

p

Figure 5: A zoom on the region of small perturbations L (the late-time region) in the spacetime described in Fig. 4.

decay, along a line of constant r, as an inverse power of t 13 (r and t are the Schwarzschild area and time coordinates, correspondingly). Along the event horizon (EH) itself, perturbations decay as an inverse power of the ingoing EddingtonFinkelstein null coordinate v 14 . We shall assume here that a similar power-law decay occurs for perturbations over a Kerr background. (This assumption is supported by recent numerical simulations 15, and by theoretical arguments 16). The weakness of perturbations at the late-time portion of the EH implies that perturbations inside the BH will be arbitrarily weak in the neighborhood of P. (More precisely, along any line of constant r, r _ < r < r +, the perturbation will vanish on the approach to P.) This is demonstrated in Ref. 16. Thus, the region of arbitrarily-weak perturbations also extends to the BH's interior, up to the CH (not necessarily including the CH itself). We shall denote this internal region of weak perturbations by L. In Fig. 5 we zoom on this region L. Note that the region L is infinitely large, as P is located in infinite geodesic distance from any point inside the BH. Since the deviations from the exact Kerr geometry are small in L, we can employ the perturbative approach to analyze the geometry inside this region. We shall now briefly outline this procedure; more details may be found in Refs.17 ,16,18. First, we express the metric as kerr + h a~ ga~ -ga~ where gkerr is the exact Kerr metric, and h is the metric perturbation. (Both these entities are tensors, but for brevity we shall often omit their indices.) We view h as a field living on the exact Kerr background. Substituting this expression in

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the vacuum Einstein equations, one obtains a closed nonlinear field equation for h (more precisely, a closed system of nonlinear field equations for the components of h). We can use this field equation to predict the evolution of h, once the initial data are given at the EH. (In principle, one also needs another set of initial data, on an ingoing null hypersurface that intersects the EH. The effect of this second set on the evolution in L is negligible in the late-time limit, however, as was shown in Ref. 16. Roughly speaking, these initial data are "infinitely red-shifted" while propagating along the EH.) Specifically, we shall assume that at the EH h behaves as v- n (like in Schwarzschild), where n is assumed to be 21 + 2, with a possible uncertainty of ±1 (however, the exact value of n is not important for our analysis). To tackle the nonlinearity of the field equation for h, we now employ the standard method of non-linear perturbation expansion. Namely, we expand h as h

= hI + h 2 + h 3 + ...

where hI is the linear metric perturbation, h 2 is the second-order metric perturbation, etc. (Hereafter, we omit the spacetime indices of h for brevity.) All fields h J satisfy a linear field equation of the schematic form (1)

Here, D is a linear differential operator (the linear piece of the vacuum Einstein equations for a perturbation over Kerr), common for all J. The source terms SJ are constructed as follows. SI vanishes identically; S2 is quadratic in hI (and its derivatives); S3 has one piece cubic in hI, and another piece proportional to the product of hI and h 2 (or their derivatives), and so on. The initial data for the field hI is taken to be that of h, whereas the fields h J with J > 1 are assumed to have zero initial data. Thus, the field h J is of Jth order in the initial data. It is essential that the source term SJ is made of terms h J' (and their derivatives) with JI < J only. This means that the equations involved in (1) may in principle be solved one at a time, and each of these equations is a linear differential equation a homogeneous equation for J = 1, and an inhomogeneous equation for J > l. In order to analyze the linear field equations enfolded in (1), we developed the method of late-time expansion. Essentially, this is an expansion in l/t (or, alternatively, l/v near the EH and l/u near the CHi u and v are the ingoing and outgoing Eddington-like coordinates for Kerr). This expansion takes advantage of the inverse power-law form of the initial data (e.g. at the EH). The method, and its application to spinning black holes, is further explained in Refs. 16,17. (The same method is demonstrated in Refs. 19,20, in full detail, for perturbations over a RN background. The full detail of its application to perturbations over Kerr will be given in Ref. 18.) In what follows we shall present the main results of the calculation. A more detailed account of these results may be found in Ref. 17. First, a few qualitative results: • All the terms h J are finite at the CH . • Moreover, all terms h J strictly vanish at the asymptotic point P (even if P is approached along the CH itself), and are therefore arbitrarily small at the early section of the CH.

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• The larger the non-linear index J, the faster the decay of h J at P. [The rate of decay of h J is roughly like (hI)J.] • Therefore, in the early portion of the CH (Le. near P), h is dominated by the linear term hI, and the contribution of all nonlinear terms is negligible. [As we shall immediately see, despite the finiteness of h at the CH, the curvature associated with it diverges there. Still, this divergence is dominated by hI at the early section of the CH, and the relative contribution of the nonlinear terms J > 1 is negligible there.] We shall now describe the main results in a more quantitative manner. We shall restrict attention here to the linear perturbation hI, as it dominates the metric perturbation (and the divergence of curvature), and for brevity we shall omit the upper index 1. As coordinates for the background Kerr geometry we shall take (U, V, e, l/J), which are closely related to those used by Chandrasekhar in Ref. 21 for the description of the (unperturbed) inner horizon of Kerr. More specifically, U and V are the tangents of Chandrasekhar's coordinates U_ and V_, respectively, l/J is the same as Chandrasekhar's

m

(here h stands for a typical metric-perturbation component, e.g. h(JIJ or huv)' The result of the calculation is [to the leading order in (In IUD- I and (In IVD- I ]

h m ~ A(e) (In IUD-n' Cos(mO In lUI)

+ B(e) (In IVI)-n" Cos(mO In IVI) .

Here, A(e) and B(e) are some regular functions of e, n' and nil are integers, n', nil ~ (n - 1), and 0 == aj2Mr _, M and a being the BH's mass and angular-momentum parameter, respectively, and r _ is the r-value of the inner horizon (for a Kerr BH with parameters M, a). One immediately observes that h m is indeed finite at the CH: hmev = 0) ~ A(e) (In IUD-n' Cos(mOln lUI) , and, moreover, it vanishes at the asymptotic past (U -+ -00). However, the derivative of hm with respect to the regular background coordinate V diverges at the CH: h m.v oc (In IVI)-n" Sin(mO In IVI) V-I -+ 00 . [for axially-symmetric modes, i.e. m

= 0, this expression is replaced by

h m •v oc (In 1V1)-(n"+I) V-I,

which again diverges at V = 0.] As a consequence, the Riemann tensor diverges at the CH: A typical Riemann component like RVIJvIJ diverges like RVIJVIJ

oc (In IVD-n" Sin(mO In IVD V- 2 -+

00 .

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[We consider here the contribution of a specific mode m, and omit the multiplicative regular dependence on () and ¢ for brevity; Note that the leading-order divergent part of Riemann is linear in (derivatives of) h l, so at the leading order the contribution of the various m-modes may be added.] The curvature scalar Riemann 2 is dominated by products of the form RVbVeRUbUe ,

where the indices b, c denote the two angular coordinates. CH, but RUbU e is finite there. One thus finds

RVbV e

diverges at the

(2)

[In this expression, we again omitted the regular dependence on U as well as on () and ¢. We considered here the contribution of a specific mode m to RVbVe, interacting with the contribution of all modes to RUbUe. Note also that Eq. (2) only holds for the nonaxially-symmetric modes. For axially-symmetric modes, the contribution is ROI{3'YoROI{hO ex: (In IVI)-(n"+1) V- 2 , which again diverges at the CH.] We conclude that the hypersurface V = 0 is the locus of a scalar-curvature singularity. [The expression (2) was originally derived from the expressions for the metric perturbations in Ref. 17. Recently, I also derived this expression by calculating the asymptotic form of the linear gravitational Newman-Penrose perturbations 'l/Jo and 'l/J4. In an appropriate gauge, one finds

The analysis of 'l/Jo and 'l/J4 is based on the separable Teukolsky equation, and is therefore much simpler than that of metric perturbations. Using the late-time expansion to find the asymptotic behavior of these two fields at the CH, one recovers Eq. (2). We emphasise that considering the linear Newman-Penrose perturbations is sufficient for our purpose, because the curvature singularity at the CH is dominated by the linear perturbations.] In the nonlinear perturbation expansion, the typical ratio of two successive terms is of the order of h (or hl, which is approximately the same). Therefore, roughly speaking, we can regard hl as the effective expansion parameter in the perturbation series. This parameter becomes arbitrarily small at (and near) the early portion of the CH (it vanishes on the approach to P), so the expansion appears to be well-behaved there. It should be emphasized, however, that at present we are unable to show, mathematically, whether the expansion converges or not. We also note that at present there still is an uncertainty of unity about the specific values of n, n', and nil, primarily due to our incomplete knowledge of the evolution of external perturbations of spinning BHs. There may also be a difference of 1 between the values of these parameters in axially-symmetric and nonaxiallysymmetric modes.

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3

Features of the CH Singularity

We shall now briefly discuss the main features of the CH singularity inside a generic spinning BH, as emerge from the perturbation analysis. • Null: First, the CH singularity is null, as follows from the following considerations: (i) In the Kruskal-like coordinates we use, g~~r and g~~rr vanish at the CH (V = 0); (ii) (hJ)uu and (hJ)Ub vanish at V = 0 for all J; (iii) as a consequence, in the perturbed spacetime, guu and gUb strictly vanish at V = 0; (iv) therefore, gVV vanishes at V = O. This means that the curvature singularity of the perturbed geometry at V = 0 is strictly null. • Weakness: As we mentioned before, the background coordinates we use are regular at the CH. Namely, in these coordinates the background metric g~iJr forms a nonsingular matrix whose components are all finite at the CH. We also saw that the metric perturbation hof3 is finite at the CH, and is also arbitrarily small at the early portion of the latter. Therefore, the full metric tensor gof3 will form a nonsingular matrix with finite components. That is, the full metric tensor is nonsingular at the CH. The singularity is therefore weak in Tipler'S terminology; Namely, the tidal deformation experienced by an infalling object will be finite as the object hits the CH, despite the divergence of curvature there. Moreover, it can be shown that the portion of the tidal deformation associated with the diverging curvature (to be distinguished from that associated with the regular Kerr background) will be arbitrarily small for a late-time infalling orbit that intersects the early portion of the CH. • Scalar-curvature: As is obvious from Eq. (2) above, the curvature scalar Riemann 2 diverges at the singular CH. • Oscillatory: It follows from Eq. (2) that if the perturbation is nonaxiallysymmetric, then, on the approach to V = 0, the curvature scalar Riemann 2 oscillates infinite number of times while diverging. The singularity at the CH is thus oscillatory. On the other hand, the axially-symmetric modes (m = 0) will yield non-oscillatory contributions to Riemann 2 • [In a generic situation, one should expect both axially-symmetric and nonaxially-symmetric modes of perturbations. Therefore, the scalar Riemann 2 will be the superposition of oscillatory and non-oscillatory components.) Note that although the explicit expressions for the metric perturbations depend on the choice of coordinates and gauge, the above mentioned features of the singularity are gauge-invariant. Obviously, the divergence of the scalar Riemann 2 is gauge-invariant, and so is the null character of the singular hypersurface. Although the specific values of the metric functions at the singular hypersurface are gauge-dependent, the notion of nonsingularity of the metric tensor is not: The very fact that there exists a gauge which regularizes the metric tensor at the singularity is itself gauge invariant. Also, the oscillatory character of the singularity may be formulated in terms of the behavior of Riemann 2 as a function of the affine parameter, along an outgoing null (or timelike) geodesic that intersects the CH singularity. This is obviously a gauge-invariant notion.

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In summary, the perturbation analysis shows that a curvature singularity forms at the CH of a generic spinning BH, and this singularity is Null, Weak, Scalarcurvature, and oscillatory. 4

Non-Perturbative Local Analytic Approach

So far, most of our knowledge about the structure and features of the CH singularity inside spinning BHs emerges from the perturbation analysis described above. This situation is somewhat unsatisfactory, especially because at present we are unable to show the convergence of the nonlinear perturbation expansion. One would therefore like to employ alternative analytic, non-perturbative, approaches, which may also provide a deeper mathematical insight into the problem. One of the factors which much complicates the analysis is the non-local character of the problem. The ultimate goal would be to analyze the evolution of "physical" initial data, defined on an initial hypersurface preceding the collapse (or, at the least, an initial hypersurface that extends to spacelike infinity), up to the formation of the spacetime singularity. The singular hypersurface, located inside the BH, is necessarily remote from the initial hypersurface, which means that a global analysis is required. The problem is much simplified if one replaces the above global setup by a local one. For example, one can consider a hypothetical initial hypersurface S which intersects the singular hypersurface. Then, local techniques may be adequate for analyzing the features of the singular hypersurface in a neighborhood of the intersection point. The main disadvantage is that, in this setup, the "initial data" on S are necessarily singular at the intersection point. Thus, with this local approach it will not be possible to show in a definitive way that a given type of singularity will evolve from regular initial data. Still, this approach can be used to answer other types of questions, concerning the local mathematical validity of any proposed ansatz of the singularity structure: 1. Is this ansatz of the singularity consistent with the field equation?

2. How generic is this ansatz? (In other words: how generic is the class of local solutions which conform with this ansatz?) Does it include a sufficient number of degrees of freedom, so as to be regarded as generic? (In this context, the "degrees of freedom" are the free functions of the three spatial coordinates involved in the local asymptotic behavior.) Clearly, a positive answer to both questions is a necessary (though not sufficient) condition for a given type of singularity to be regarded as "physical". (If a singularity fails to be generic, then it cannot be stable in a global setup.) Motivated by these considerations, Flanagan and myself recently carried out such a local analysis 22. We used the Cauchy-Kowalewski theorem to construct, mathematically, a class of local vacuum solutions, which all admit a null weak singularity. The "initial functions" in this construction are defined on a spacelike hypersurface, which intersects the null singular hypersurface on a spacelike twosurface. These initial functions are functions of the three spatial coordinates, which

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do not admit any symmetry and are quite generic, apart from the demand for analyticity (the analyticity is required in order for the Cauchy-Kowalewski theorem to be applicable). We denote our coordinates by x, y, u, v, and we use the gauge condition guo; = guy = guu = gvv = 0 in our construction. The class of solutions we constructed depends on eight freely-specifiable (analytic) functions of the three spatial coordinates; And, in the gauge we use, the generic solution of the vacuum Einstein equations also depends on eight such freely-specifiable functions. Our construction thus shows that from the local point of view, the null weak singularities are not only consistent with the vacuum Einstein equations, but are also generic. The construction also implies that generically these null weak singularities are scalar-curvature. In Ref. 23 I apply the local approach to the class of (linearly-polarized) two colliding plane waves spacetimes. Again, I show that null weak scalar-curvature singularities are locally-generic within this class of spacetimes. The extent of this result is limited compared to the analysis in Ref. 22, because it is limited to planesymmetric spacetimes. Yet, it has the advantage that the initial functions need not be analytic (the construction in Ref. 23 does not relay on the Cauchy-Kowalewski theorem). In a previous work, Brady and Chambers 24 used a similar approach to analyze the local neighborhood of the singularity: They used the 2+2 formulation 25 of the Einstein equations and constructed sets of initial functions on an outgoing null hypersurface (viewed as characteristic initial data), which admit a curvature singularity at some spacelike two-surface. They demonstrated the existence of a generic class of such singular initial-data sets which solve the constraint equations (formulated on the outgoing null hypersurface). If one makes the assumption that an (ingoing) null singularity will always evolve from such a singular two-surface, then this null singularity will be generic. They also showed that the formation of such an ingoing null singularity is itself consistent with the constraint equations. Note, however, that the consistency ofthe construction with the Einstein's evolution equations was not addressed in 24. This analysis does not determine, therefore, what type of singularity will evolve from the initial singular two-surface (e.g. whether it will be spacelike or nUll). The more recent analyses 22,23 take care of the full set of Einstein equations - both the constraint and evolution equations. 5

Numerical Simulations

It may be tempting to try verify the above picture of the CH singularity numerically. The ultimate goal would be to evolve numerically the fully nonlinear set of (vacuum) Einstein equations (starting from initial data corresponding to the Kerr geometry plus generic initial perturbations), all the way from the initial hypersurface and up to the formation of a curvature singularity. At present, however, the numerical simulation in 3+1 (and perhaps even 2+1) non-trivial dimensions, combined with the special complications of our problem ( e.g. the necessity to approach the curvature singularity), is obviously two ambitious. For this reason, the numerical attempts were so far restricted to the toy model of a spherical charged black hole. (See, however, the recent analyses by Krivan et al l5 of linear perturbations outside a Kerr

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BH.) In particular, in the last few years several researchers numerically analyzed the model of a spherical charged BH nonlinearly perturbed by a self-gravitating massless scalar field. I will now very briefly mention these analyses and their main results. The first numerical simulation of this model was carried out by Gnedin and Gnedin 26. In this simulation, however, no much attention was paid to the neighborhood of the CH, and, moreover, the code was "badly behaved" there (as the authors themselves point out; In fact, a close look reveals that the non-standard null coordinates and grid used in 26 are not suitable for a numerical study of the region near the CH: With these coordinates and mesh, even if one uses a very large grid, say, 10000X10000, there are no grid points in the relevant range Veddington » M.) Later, a numerical simulation by Brady and Smith 27 confirmed that indeed a null curvature singularity forms at the CH, were the area coordinate r is finite (but decreases monotonically with increasing affine parameter along the CH) and the mass function diverges (mass-inflation), indicating a scalar-curvature singularity. (They also found that when r shrinks to zero, a spacelike singularity forms. It is not obvious yet whether this spacelike singularity of the spherical scalar field is relevant to spinning vacuum BHs.) Recently, Burko 28 performed a numerical simulation of the same model. He recovered the above results by Brady and Smith, and also confirmed several additional aspects of the above perturbative picture of the CH singularity: First, the metric tensor (in appropriate, "Kruskal-like", double-null coordinates U, V) is regular at the CH, indicating a weak singularity. Second, at the early portion of the CH the metric functions were found to deviate only slightly from those of the pure RN geometry (despite the divergence of curvature there). Moreover, the perturbation in the metric function r(U, V) (i.e. the deviation from the corresponding function r(U, V) in pure RN) is well described by the leading order in the nonlinear perturbation series. In addition, the evolution of the scalar field is also well described by the linear perturbation analysis 6,19,20. This confirms (in the spherical case) the above predictions of the perturbative approach (for a spinning BH): At the early portion of the CH, the deviations of the metric functions from those of the unperturbed stationary background geometry are arbitrarily small, and are, moreover, well described by the linear perturbation, i.e. the leading-order term in the nonlinear perturbation expansion. [It should be pointed out that the spherically-symmetric toy model considered in this section cannot be used to verify the oscillatory character of the CH singularity of realistic spinning BHs: Since the parameter n in Eq. (2) is proportional to the spin parameter a, it vanishes in all spherically-symmetric toy models.] 1. J. M. Bardeen, Nature 226, 64 (1970). 2. K. S. Thorne, Astrophys. J. 191, 507 (1974). 3. V. de la Cruz and W. Israel, Nuovo Cimento 51A, 744 (1967) 4. R. R. Penrose, in Battelle Rencontres, 1967 lectures in mathematics and physics, edited by C. M. DeWitt and J. A. Wheeler (Benjamin, New York, 1968), p. 222 . 5. M. Simpson and R. Penrose, Inter. J. Theor. Phys. 7, 183 (1973).

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6. Y. Giirsel, V. D. Sandberg, I. D Novikov and A. A. Starobinsky, Phys. Rev. D 19, 413 (1979); Y. Giirsel, I. D Novikov, V. D. Sandberg and A. A. Starobinsky, Phys. Rev. D 20, 1260 (1979). 7. S. Chandrasekhar and J.B. Hartle, Proc. R. Soc. Lond. A 384,301 (1982). 8. I. D. Novikov and A. A. Starobinsky, Abstract of contributed papers of the 9th Intern. Conf. on General Relativity and Gravitation, Jena, DDR, p. 268 (1980) . 9. W. A. Hiscock, Phys. Lett. A 83, 110 (1981). 10. E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990). 11. A. Ori, Phys. Rev. Lett. 67, 789 (1991). 12. F. J. Tipler, Phys. Lett. A 64, 8 (1977). 13. R. H. Price, Phys. Rev. D 5, 2419 (1972); ibid 2439 (1972). 14. C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D 49, 883 (1994). 15. W. Krivan, P. Laguna, and P. Papadopoulos, and N. Anderson, Phys. Rev. D 54, 4728 (1996); W. Krivan, P. Laguna, P. Papadopoulos, and N. Anderson, Phys. Rev. D 56, 3395 (1997) 16. A. Ori, Gen. Rel. Grav. 29, 881 (1997). 17. A. Ori, Phys. Rev. Lett. 68,2117 (1992). 18. A. Ori, in preparation. 19. A. Ori, Phys. Rev. D 55,4860 (1997). 20. A. Ori, Phys. Rev. D (in press) (gr-qc/971134). 21. S. Chandrasekhar, The mathematical theory of black holes (Oxford University press, 1983) §58. 22, A. Ori, and E. E. Flanagan, Phys. Rev. D 53, R1754 (1996). 23. A. Ori, "Null weak singularities in plane-symmetric spacetimes", submitted to Phys. Rev. D. 24. P. R. Brady and C. M. Chambers, Phys. Rev. D 51, 4177 (1995). 25. S. A. Hayward, Class. Quantum Grav. 10, 773 (1993), ibid 779 (1993). 26. M.L. Gnedin and N.Y. Gnedin, Class. Quantum Grav. 10, 1083 (1993). 27. P.R. Brady and J.D. Smith, Phys. Rev. Lett. 75, 1256 (1995). 28. L.M. Burko, Phys. Rev. Lett. (in press) (gr-qc/9710112).

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HAS THE BLACK HOLE EQUILIBRIUM PROBLEM BEEN SOLVED? B. CARTER D.A.R.C., (UPR 176, CNRS), Observatoire de Paris, 92 Meudon, France When the term "black hole" was originally coined in 1968, it was immediately conjectured that the only pure vacuum equilibrium states were those of the Kerr family. Efforts to confirm this made rapid progress during the "classical phase" from 1968 to 1975, and some gaps in the argument have been closed during more recent years. However the presently available demonstration is still subject to undesirably restrictive assumptions such as non-degeneracy of the horizon, as well as analyticity and causality in the exterior.

1

Introduction

The purpose of this report is to present a brief overview of the present state of progress on the still not completely solved problem of the classification of black hole equilibrium states within the (astrophysically motivated) context of Einstein's pure vacuum theory and its electrovac generalisation in ordinary four dimensional spacetime. In recent years there has been a considerable resurgence of mathematical work on black hole equilibrium states, but most of it has been concerned with more or less exotic generalisations, not restricted to four dimensions, involving speculative extensions of Einstein's theory to allow for inclusion of various scalar and other (e.g. Yang Mills type) fields such as those ocurring in low energy limits of superstring theory. The present review will not even attempt to deal with this rapidly developing and open ended area of investigation. The not quite so fashionable but - as far as the observable physical world is concerned - more soundly motivated subject of black hole equilibrium states with surrounding rings of matter (such as would result from accretion from external sources) is also beyond the scope this article. One reason why there has not been so much recent work on what, from an astrophysical point of view, is the most important problem in black hole equilibrium theory, namely that of the pure vacuum states in four dimensions, is the widespread belief that the problem was solved long ago, and that the solution is just what was predicted by my original 1967 conjecture 1,2, i.e. that it is completely provided by the subset of Kerr solutions 3 for which a 2 ~ M2 where a = J / M is the ratio of the angular momentum J to the mass M. This belief rapidly gained general acceptance in astrophysical circles when - following the example of Israel's earlier 1967 work 4 providing strong evidence that (as has since been confirmed) the only strictly static (not just stationary) solutions were given by the special Schwarzschild (J = 0) case - I was able 5 in 1971 to obtain a line of argument that provided rather overwhelming, though by no means absolutely watertight, mathematical evidence to the effect that the most general solution is indeed included in the Kerr family. Many of the interested parties, particularly observationally motivated astrophysicists, considered that the conversion of the original plausibility argument into an utterly unassailable mathematical proof was merely a physically insignificant

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technical formality, that could be left as an exercise for the amusement of obsessively rigoristic pure mathematicians. However such lack of interest was not the only reason why subsequent progress on the problem has been rather slow. It was soon shown by those - starting with Hawking 6 ,7 - who took the question seriously after all that the mathematical work needed to deal with the various apparently small technical gaps and loose ends in the comparitively simple 1971 argument 5,8 is harder than might have been hoped. Thus despite the very considerable efforts of many people - of whom some of the most notable after Hawking 6 ,7,9 have been Robinson 10,11,12, followed (for the electrovac generalisation) by Bunting 13,14 and Mazur 15,16), and most recently (since my last comprehensive review of the subject 17) Wald and his collaborators 18,20,19,21 - there still remains a lot that needs to be done before we shall have what could be considered a mathematically definitive solution even of the pure vacuum problem, not to mention the more formidable challenge of its electrovac generalisation. It will be convenient to present the results in chronological order, which roughly corresponds to that of their logical development except for a few cases in which newer work has provided more elegant methods of rederiving results that were orginally obtained by more laborious means. Section 2 rapidly recalls some of the relevant results (culminating in Israel's theorem) obtained during the prolonged period of general confusion that I refer to as the "preclassical phase", prior to the introduction of the term "black hole" and to the general recognition that the disciples of Ginzburg and Zel'dovich had been right 22 in arguing that what it represents is a generic phenomenon - not just an unstable artefact of spherical symmetry as many people, including Israel himself 23, had speculated. Section 3 describes the rapid progress made during what I refer to as the "classical phase" - the beginning of what Israel 23 has referred to as the "age of enlightenment"immediately following the definitive formulation of the concept of a "black hole" (in terms of the "outer past event horizon" in an asymptotically flat background) so that the corresponding equilibrium state problem could at last be posed in a mathematically well defined form. Section 4 describes the substantial though slower progress that has been made in what I refer to as the "postclassical phase", that began when the main stream of work on black holes had been diverted to quantum aspects following the discovery of the Hawking effect 24. The final section draws the intention of newcomers to the field, for whom this article is primarily intended, to the mathematically challenging (even if physically less important) problems that still remain to be tackled: these include not only the questions concerning the technicaly awkward degenerate limit case and the assumptions about spherical topology and analyticity that have been discussed by Chrusciel 25 and also, in a very extensive and up to date review, by Heusler 26, but also the largely neglected question of the assumption of causality, i.e. the absence of closed timelike curves. A propos of the latter, is ironic that while providing a fascinating account of the mental blockages that impeded earlier workers in the theory of both black holes and time machines (i.e. regions of spacetime threaded by closed timelike curves) Thorne's recent history of "Black holes and time warps" 22 has a blind spot of its own, in that it discusses only the kind of closed timelike curve whose presence

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depends on topological multiconnectedness in "wormholes" of a rather artificial kind (so that the resulting causality violation is "trivial" 27 in the sense of being in principle removable by replacing the spacetime model by its locally equivalent universal covering space). What is rather surprising is Thorne's failure to mention the kind of time machine exemplified 2 by the Kerr solutions for a 2 > M2, in which causality violation of a more "flagrant" 27 (not so easily removable) kind occurs. In the Kerr black hole case a2 ~ M2 the causality violation is confined to the interior, but the unsolved problem is whether there exist other black hole equilibrium solutions in which such causality violation occurs outside the horizon. The formal existence of such pathological black hole solutions might of course be reasonably supposed to be irrelevant for realistic physical purposes. However the same kind of objection could be raised to Thorne's "wormhole" time machines: if the latter are nevertheless at least of sufficient mathematical interest to be worth investigating then the same applies a fortiori to black hole time machines if they exist, a possibility that is by no means excluded by any of the work carried out so far. 2

The ("unenlightenned") prec1assical phase (1915-67)

What I refer to as the preclassical phase in the development of black hole theory is the period of unsystematic accumulation of more or less relevant results prior to the actual use of the term "black hole". This period began with the discovery in 1916 by Schwarzschild 28 of his famous asymptotically flat vacuum solution, whose outer region is strictly static, with a hypersurface orthogonal timelike Killing vector whose striking feature is that its magnitude tends to zero on what was at first interpreted as a spacetime singularity, but was later recognised to be interpretable as a regular boundary admitting a smooth extension to an inner region where the Killing vector becomes spacelike. This preclassical phase culminated in Israel's 1967 discovery 4 of a mathematical argument to the effect that the Schwarzschild solution is uniquely characterised by these particUlar properties, i.e. the original spherical example is the only example. The significance of this discovery was the subject of an intense debate that precipitated the transition to the "classical phase" , inaugurating what Israel 23 has termed the "age of enlightenment" , which dawned when the preceeding confusion at last gave way to a clear concensus. Thrilling eyewittness accounts of the turbulent evolution of ideas in the "golden age" of rapid progress, during the transition from the preclassical to the classical phase, have been given by Israel himself 23 and from a different point of view by Thorne 22 , while a historical account of the more dilatory fumbling in the early years of the preclassical phase has been given by Eisenstaedt 29 . During most of the "unenlightenned" preclassical period, from 1915 until about 1960, nearly all the relevant work, starting with that of Schwarzschild, was in fact based on the simplifying postulate of spherical symmetry. An important consequence of this restriction was demonstrated by Birkhoff's 1923 theorem 30, which showed that the staticity property used in Schwarzschild's derivation of his solution need not have been postulated indendently of the spherical symmetry, since it followed as an automatic consequence of the vacuum field equations. An important

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step towards the concept of what would be called a "black hole" was the analysis 31 of the gravitational collapse of pressure free matter by Oppenheimer and Snyder in 1939. However in the pure vacuum case, on which the present review is focussed, progress stayed remarkably slow for a very long time, and people remained confused by the special limit in the Schwarzschild solution where the circumferencial radius r reaches the value 2M (in relativistic units). Despite the construction of analytic extensions beyond this limit by many earlier workers 32,33,34,35,36,37 a clear understanding was obtained only after more complete extensions were made by Frondsal 38, Kruzkal 39, and Szekeres 40 . Due to a renaissance of interest (following the observational discovery of quasars) progress was much more rapid during what Thorne 22 has referred to as the "golden age", which began during the last half dozen years of the preclassical phase and continued through what I call the classical phase (ending when most of the easiest problems had been solved at about the time Hawking diverted attention attention to less astrophysically relevant quantum effects). It was during the late "golden" period of the pre-classical phase, roughly from 1961 to 1967, that results of importance for vacuum black hole theory began to be obtained without reliance on a presupposition of spherical symmetry. The most important of these results were of course Kerr's 1963 discovery 3 of the family of stationary asymptotically flat vacuum solutions characterised by a degenerate "type D" Weyl tensor, and the 1967 Israel theorem 4 referred to above. As well as these two specific discoveries, the most significant development during this final "golden" period of the preclassical phase was the animated debate under the leadership of Ginzburg and Zel'dovich 41 in what was then the Soviet Union, of Wheeler and later on Thorne 42 in the United States, and of Sciama and Penrose 43 in Britain - from which the definitive conceptual machinery and technical jargon of black hole theory finally emerged. Prior to the discovery of the Kerr solution 3, when the only example considered was that of Schwarzschild, it had not been thought necessary to distinguish what Wheeler later termed the "ergosphere" - where the Killing vector generating the stationary symmetry of the exterior ceases to be timelike - from the "outer past event horizon" bounding what Wheeler later termed the "black hole" region, from which no future timelike trajectory can escape to the asymptotically flat exterior. In its original version, Israel's 1967 theorem 4 (as well as its electrovac generalisation 45) was effectively formulated in terms of an "infinite redshift surface" that was effectively taken to be what in strict terminology was really the "ergosphere" rather than the "outer past event horizon": this meant that the significance of the theorem for the theory for what were to be called a "black holes" was not clear until it was understood that (as shown in my thesis 1,27 and pointed out independently by Vishveshwara 44) subject to the condition of strict staticity postulated by Israel (but not in the more general stationary case exemplified by the non-spherical Kerr 3 solutions) the the "outer past event horizon" actually will coincide with the ergosphere. One of the first to appreciate the distinction between (what would come to be known as) the horizon and the ergosphere, and to recognise the members of the relevant (a 2 ~ M2) Kerr subset as prototypes of what would come to be known as black hole solutions was Boyer 46,47,48 . However at the time ofthe first detailed ge-

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ometrical investigations of the Kerr solutions 49,47 (the purpose for which, following a suggestion by Penrose, I originally introduced the scheme of representation by the kind of conformal projection now commonly known as a "Penrose diagram"), it was assumed by Boyer and the others involved, including myself, that we were dealing just with a particularly simple case within what might turn out to be a much more extensive category. However the publication of the 1967 Israel theorem 4 - which went much of the way towards proving that the Schwarzschild 28 solution is the only strictly static example - immediately lead to the question of whether the Kerr solutions might not be similarly unique. The explicit formulation of this suggestion came later to be loosely referred to in the singular as the "Israel-Carter conjecture", but there were originally not one but two distinct versions. The stronger version - suggested by the manner in which the Israel theorem was originally formulated - conjectured that the relevant Kerr subfamily might be the only stationary solutions that are well behaved outside and on a regular "infinite redshift surface" - a potentially ambiguous term that in the context of the original version 4 of Israel's theorem effectively meant what was later to be termed an "ergosurface" rather than an "event horizon". The weaker version, first written unambiguously in my 1967 thesis 1,2, conjectured that the relevant Kerr subfamily might be the only stationary solutions that are well behaved all the way in to a regular black hole horizon, not just outside the ergosphere. Work by Bardeen 50 and others on the effects of stationary orbitting matter rings (which can occur outside the horizon but inside the ergosphere of an approximately Kerr background) soon made it evident that strong version of the conjecture is definitely wrong, no matter how liberally one interprets the rather vague qualifications "regular" and "well behaved". On the other hand the upshot of the work to be described in the following sections is to confirm the validity of my weaker version, as expressed in terms of the horizon rather than the ergosphere. It is nevertheless to be remarked that, as was pointed out by Hartle and Hawking 51, the generalisation of this conjecture from the Kerr pure vacuum solutions 3 to the Kerr-Newman electrovac solutions 52 is not valid, since the solutions due to Papapetrou 53 and Majumdar 54 provide counterexamples. It is also to be emphasised that, as will be discussed in the final section, the question still remains entirely open, even in the pure vacuum case, if the interpretation of the qualification "well behaved" is relaxed so as to permit causality violation outside the horizon of the kind that is actually observed 2 to occur in the inner regions of the Kerr examples. 3

The classical phase (1968-75)

What I refer to as the classical phase in the development of black hole theory began when the appropriate conceptual framework and the corresponding generally accepted technical terminology became available, facilitating clear formulation of the relevant mathematical problems, whose solutions could then be sought by systematic research programs, not just by haphazard approach of the preclassical period. The relevant notions had already began to become clear to a small number of specialists (notably Wheeler's associates, including Thorne and Misner, in the United States, and Penrose's associates, including Hawking and myself in Britain) during

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the period of accelerated activity at the end of what I call the preclassical phase. However it was not just theoretical progress that precipitated the rather sudden ("first order") transition to what I call the classical phase. Just as it was the discovery of the quasar phenomenon that stimulated the "golden age" 22 of rapid progress, so also, rather similarly, it was another observational event, namely the accidental discovery of pulsars by Bell and Hewish, that inaugurated the transition from the "preclassical phase" to the "age of enlightenment" 23 during the second half ofthe "golden age". Unlike the quasar phenomenon, whose underlying mechanism is far from clear even to day, the pulsar phenomen was rapidly elucidated: in the early months of 1968 it was already generally generally recognised to be attributable to neutron stars, whose likely existence had long been predicted by theoreticians, but whose reality had never until then been taken very seriously by the majority of the astronomical community. The 1968 confirmation that neutron stars definitely exist and are directly observable immediately transformed the status of theoreticiens in the eyes of the observers. (Prior to 1968 even our firmest affirmations were treated with the greatest scepticism; after 1968 even our most tentative speculations, as well as our conjectures about "black holes" , were received as oracular pronouncements.) This meant that the beginning of the "classical phase" was characterised not only by the establishment of an "enlightenned" consensus among the previously disparate groups of specialists working in the field, but also by the recognition for the first time by a much wider public that a new and important field of theoretical astrophysics had been born. At the beginning of 1968 the term "black hole" was known only to a handful of participants in the seminars organised at Princeton by Wheeler; by the end of 1968 the term had already been widely publicised in televised science fiction so that it was already known (if not understood) by millions of people allover the world. At a time when the existence of black holes produced by burnt out stars throughout our galaxy and others was already widely albeit prematurely recognised by much of the astronomical community, it became urgent for the theoreticians actually working in the field to settle the question of the physical relevance of the black hole scenario, which requires that it should occur not just as an unstable special space (which was the implication that Israel was at first inclined to draw from his theorem 23) but as a generic phenomenon as Zel'dovich and his collaborators had been claiming22). The strongest conceivable confirmation of the general validity of the black hole scenario is what would hold if Penrose's 1969 cosmic censorship conjecture 55 were valid in some form. According to this vaguely worded conjecture, in the framework of a "realistic" theory of matter the singularities resulting (according to Penrose's earlier "preclassical" closed trapped surface theorem 43) from gravitational collapse should generally be hidden within the horizon of a black hole with a regular exterior. However far from providing a satisfactory general proof of this conjecture, subsequent work on the question (of which there has not been as much as would be warranted) has tended to show that can be valid only if interpreted in a rather restricted manner. Nevertheless, despite the construction of various more or less artificial counterexamples by Eardley, Smarr, Christodoulou and others 56 to at least the broader interpretations of this conjecture, it seems clear that there will

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remain an extensive range of "realistic" circumstances under which the formation of a regular black hole configuration is after all to be expected. It remains a controversial question (and in any case one that is beyond the scope of this discussion of pure vacuum equilibrium states) just how broad a range of circumstances can lead to regular black hole formation, and whether or not "naked singularities" can sometimes be formed instead under "realistic" conditions. However that may be, all that is actually needed to establish the relevance of black hole for practical physical purposes (as a crucial test of Einstein's theory, and assuming the result is positive, as an indispensible branch of astrophysical theory) is the demonstration of effective stability with respect to small purturbations of at lease some example. This essential step was first achieved in a mathematically satisfactory manner for the special case of the original prototype black pole solution, namely the Schwarzschild solution, in a crucially important quasi-normal mode analysis 57 by Vishveshwara in 1970. Another important article 58 by Price provided a more detailed account of the rate at which the solution could be expected to tend towards the Schwrzschild form under realistic circumstances as seen from the point of view of an external observer. The work of Vishveshwara and Price put the physical relevance of the subject beyond reasonable doubt by demonstrating that this particular (spherical) example is not just stable in principle but that it will also be stable in the practical sense of tending to its stationary (in this particular case actually static) limit within a timescale that is reasonably short compared with other relevant processes: in the Schwarzschild case the relevant timescale for convergence, at a given order of magnitude of the radial distance from the hole, turns out just to be comparable with the corresponding light crossing timescale. During the remainder of the "classical phase", important work 59,60,61) by Teukolsky, Press and others made substantial progress towards the confirmation that the Kerr solutions are all similarly stable so long as the specific angular momentum parameter a = JIM is less than its maximum value, a = M. However the possibility that instability might set in at some intermediate value in the range o < a < M was not conclusively eliminated until much later on, in the "postclassical" era, when (following a deeper study of the problem by Kay and Wald 62) the question was settled more conclusively by the publication of a powerful new method of analysis developed by Whiting 63. While this work on the stability question was going on, one of the main activities characterising the "classical phase" of the subject was the systematic investigation (along lines pionneered 64,65 by Christodoulou and Ruffini) of the general mechanical laws governing the benaviour of stationary and almost stationary black hole states. Work by a number of people including Hawking, Hartle, Bardeen, and myself 6 ,66,67,68,69 (and later, as far as the electromagnetic aspects 8,70,71,72 are concerned, also Znajek and Damour) revealed a strong analogy with the thermodynamical behaviour of a viscous (and electrically resistive) fluid. (Following a boldly imaginative suggestion by Bekenstein 73, the suspicion that this analogy could be interpreted in terms of a deeper statistical mechanical reality was spectacularly confirmed 24 when Hawking laid the foundations of quantum black hole theory.) It was the substantial theoretical framework built up in the way during the "classical" phase that decisively confirmed the crucial importance of the equilib-

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rium state problem on which the present article is focussed. Returning to this more specialised topic, I would start by recalling that - as well as the provision of the first convincing demonstration that (contrary to what Israel 23 had at first been incined to suspect) a black hole equilibrium state can indeed be stable - a noteworthy byproduct of Vishveshwara's epoch making paper 57 was its analysis of stationary as well as dynamical perturbations, which provided evidence favorable to my uniqueness conjecture 1,2 in the form of a restricted "no hair" theorem to the effect that the only stationary pure vacuum generalisations obtainable from a Schwarzschild black hole by infinitesimal parameter variations are those of the Kerr family (in which relevant small parameter is the angular momentum J). Encouraged by Vishveshwara's confirmation 57 of the importance of the problem, I immediately undertook the first systematic attempt 5,8 at verification of my uniqueness conjecture for the a 2 ~ M2 Kerr solutions as stationary black hole states. For the sake of mathematical simplicity I restricted my attention to the case characterised by spherical topology and axial symmetry, conditions that could plausibly be guessed to be mathematically necessary in any case. I also ruled out consideration of conceivable cases in which closed timelike or null lines occur outside the black hole horizon (not just inside as in in the Kerr case 2 ), a condition that is evidently natural on physical grounds, but that is not at all obviously justifiable as a mathematically necessity. Within this framework I was able in 1971 to make two decisive steps forward, at least for the generic case for which there is a non zero value of the decay parameter K, which (in accordance with the "zeroth" law of black hole thermodynamics 69) must always be constant over the horizon in a stationary state. The first of these steps 5 was the reduction of the four dimensional vacuum black hole equilibrium problem to a two dimensional non-linear elliptic boundary problem, for which the relevant boundary conditions involve just two free parameters: the outer boundary conditions depend just on the mass M and the inner boundary conditions depend just on the horizon scale parameter, c which is proportional to the product of the decay parameter K, with the horizon area A. The precise specification of this parameter c (originally denoted by the letter b, and commonly denoted in more recent litterature by the alternative letter p,) is given by the definition c = K,A/ 411', and its value in the particular case of the Kerr black holes is given the formula c = J M2 - a 2 with a = J / M. The second decisive step obtained in 1971 was the demonstration 5 that the two dimensional boundary problem provided by the first step is subject to a "no hair" (Le. no bifurcation) theorem to the effect that within a continuously differentiable family of solutions (such as the Kerr family) variation between neighbouring members is fully determined just by the corresponding variation of the pair of boundary value parameters, Le. the solutions belong to disjoint 2-parameter families in which the individual members are fully specified just by the relevant values of M and c. The only known example of such a family was the Kerr solution, which of course includes the only spherical limit case, namely that of Schwarzschild. The theorem therefore implied that if, contrary to my conjecture, some other non-Kerr family of solutions existed after all, then it would have the strange property of being unable to be continuously varied to a non-rotating spherical limit. On the basis of expe-

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rience with other equilibrium problems this strongly suggested that, even if other families did exist, they would be unstable and therefore physically irrelevant, unlike the Kerr solutions which, by Visveshwara's work 57 were already known to be stable at least in the neighbourhood of the non-rotating limit. Having drawn the conclusion from this plausible but debateable line of argument that for practical astrophysical purposes a pure vacuum black hole equilibrium could indeed be safely presumed to be described by a Kerr solution, I turned my attention to the problem of generalising this argument from the pure vacuum to the electrovac case. In the degenerate (I\: = 0) case, it had been pointed out by Hartle and Hawking 51 that the Kerr-Newman 52 family did not provide the most general equilibrium solution, due to the existence of counterexamples provided by the Papapetrou-Majumdar solutions, but it remains plausible to conjecture that the most general non-degenerate solutions are indeed provided by the Kerr-Newman family (whose simple spherical limit is the Reissner-Nordstrom solution). The electrovac generalisation of the first step of my 1971 argument 5 turned out to be obtainable without much difficulty 8, the only difference being that the ensuing two-dimensional non-linear boundary problem now involved two extra parameters representing electric charge and magnetic monopole. As in the pure vacuum case, an essential trick was the use of a modified Ernst 74 transformation based on the axial Killing vector (instead of the usual time translation generator) so as to obtain a variational formulation for which - assuming causality - the action would be positive definite. The second step was more difficult: I did not succeed in constructing a suitable electrovac generalisation of the divergence identity - with a rather complicated but (like the action) positive definite right hand side - that had enabled me to establish the pure vacuum "no hair" theorem 5 for axisymmetric black holes in 1971, but an electrovac identity of the required - though even more complicated - form was finally obtained by Robinson 10 in 1974. While this work on the electromagnetic generalisation was going on, a deeper investigation of the underlying assumptions was initiated by Hawking 6 ,7,9, who made very important progress towards confirmation of the supposition that the topology would be spherical, and that the geometry would be axisymmetric. The latter was achieved by I call the "strong rigidity" theorem, which was originally advertised 9 as a demonstration that - assuming analyticity - the black hole equilibrium states would indeed have to be axisymmetric (and hence by my earlier "weak rigidity" theorem 66 uniformly rotating) except in the static case, which in the absence of external matter was already known - from the recent completion 75,76 of the program initiated by Israel 4 ,45 - to be not just axisymmetric but geometrically (not just topologically) spherical. The claim to have adequately confirmed the property ofaxisymmetry 9 was however one of several exagerations and overstatements that were too hastily put forward during that exciting "classical" period of breathlessly rapid progress. In reality, all that was mathematically established by the "strong rigidity" theorem was just that in the non axisymmetric case the equilibrium state would have to be "non-rotating" (in the technical sense that is explained in the appendix). The argument to the effect that this implied staticity depended on Hawking's generalisation 6 of the original Lichnerowicz 77 staticity theorem, which in turn assumed the

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absence of an "ergosphere" outside the (stationary non-rotating) horizon - a litigious supposition whose purported justification in the non-rotating case was based on heuristic considerations 9 that have since been recognised to be fundamentally misleading, due to the existence of counterexamples. A satisfactory demonstration that the non-rotating case must after all be static was not obtained until the comparitively recent development 18,19,20 (described in next section) of a new and much more effective approach (allong lines summarised in the appendix) that was initiated by Wald in the more serene "post classical" era. A similarly overhasty announcement of my own during the hurry of the "classical phase" was the claim 8 to have obtained an electromagnetic generalisation of Hawking's Lichneroicz type of staticity theorem using just the same litigious assumption (which fails anyway for the rotating case) of the absence of an ergosphere, i.e. strict positivity, V > 0, of the effective gravitational potential defined as the norm, V = -k"k", of the stationarity generator. What was shown later by a more careful analysis 17 was that (after correction of a sign error in the original version 8) an even stronger and more highly litigious inequality was in fact required - until it was made finally redundant by the more effective treatment recently developed by Wald and his associates 18,19,20 on the lines summarised in the appendix. (In dealing with the related "circularity" theorem, on which the treatment 5,8 of the stationary case depends, I was more fortunate: my electromagnetic generalisation 80 of Papapetrou's pure vacuum prototype 81 has stood the test of time). Among the other noteworthy overstatements from the hastily progressing "classical" period, a particularly relevant example is Wald's own premature claim 78 (based on what turns out to have been an essentially circular argument) to have gone beyond my 1971 "no hair" theorem 5 to get a more powerful uniqueness theorem of the kind that was not genuinely obtained until Robinson's 1975 generalisation from infinitesimal to finite differences of the divergence identity I had used. In achieving this ultimate tour de force Robinson 11 effectively strengthenned the "no hair" theorem to a complete uniqueness theorem, thereby definitively excluding the - until then conceivable - existence of a presumably unstable non-Kerr branch of topologically spherical axisymmetric causally well behaved black hole solutions. Having already succeeded in generalising my original infinitesimal divergence identity 5 from the pure vacuum to the electrovac case 10, Robinson went on to try to find an analogous generalision of his more powerful finite difference divergence identity 11 from the pure vacuum to the electrovac case. However this turned out to be too difficult, even for him, at least by the unsystematic, trial and error, search strategy that he and I had been using until then. As I guessed in a subsequent review 72, there was "a deep but essentially simple reason why the identities found so far should exist" and "the generalisation required to tie up the problem completely will not be constructed until after the discovery of such an expanation, which would presumably show one how to construct the required identities directly". It was only at a later stage, in the "post classical" period that, as Heusler 26 put it " this prediction was shown to be true" when, on the basis of a deeper understanding, such direct construction methods were indeed obtained by Mazur 15,16 and, independently, using a different (less specialised) approach, by Bunting 13,14.

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4

The post-classical phase (since 1975)

Robinson's 1975 discovery of the finite-difference divergence identity 11 marked the end of what I call the "classical phase", whose focal event had been the 1972 Les Houches summer school 7,8,50 at which all the various aspects of black hole theory were assembled and treated together for the first and probably the last time. Since 1975 the subject has split into mutually non-interacting branches. On one hand there has been the new subject of quantum black hole theory: the discovery of the Hawking effect 24 aroused interest in the possible occurrence in the early universe of microscopic black holes for which such effects might be important, and this in turn lead to an interest into the conceivable effects (e.g. as potential contributors of black hole "hair") of various kinds of exotic (e.g. Yang Mill or dilatonic) fields that might have been relevant then. On the other hand, eschewing such rather wild speculations in favor of what more obviously exists in the real world, astrophysicts have been mainly interested in macroscopic black holes (of stellar mass and upwards) for which the only relevant long range interaction fields are still believed to be just gravitation and electromagnetism, but for which local mechanisms such as accreting plasma can produce spectacular effects that are thought to be responsible for many observed phenomena ranging in scale and distance from quasars down to galactic X-ray sources such as Cygnus X-I. After the development of these disconnected branches, work on the pure vacuum black hole equilibrium problem and its electrovac generalisation proceeded rather slowly. Starting with Bekenstein 79, most quantum black hole theorists were more concerned about generalising the problem to hypothetical fields of various new (e.g. dilatonic types), whereas most astrophysical black hole theorists were concerned just with accreting matter that could be treated as a small perturbation on a pure vacuum background, whose equilibrium states they supposed to have been definitely established to consist just of the relevant (a ~ M) subfamily. Only a handful of mathematically oriented theorists remained acutely aware that the definitive establishment of this naive supposition was still not complete. Another reason why progress in the theory of vacuum equilibrium states slowed down in the "post classical phase" was that the problems that had been solved in the "classical phase" had of course tended to be those that were easiest. In so far as the equilibrium problem is concerned, the most salient developments in the earlier "post classical" years were the completion referred to above by Mazur 15,16 and Bunting 13 ,14 of my work 5,8 and Robinson's 10,11 on the axisymmetric case, and the completion and streamlining 12,82,83 by Robinson, Simon, Bunting and Massood-ul-alam of the work 4,45,75,76 initiated by Israel on the strictly static case. Unlike the work just cited, which built upwards from the (not always entirely reliable) basis established in the classical period 5,6,8,9,75,76, a more recent resurgence of activity 18,20,19,21,25,84 - initiated by Wald and continued most recently by Chrusciel - has been more concerned with treating the shaky elements in the foundations of that underlying basis itself. This work has successfully closed an outstanding loophole in the previous line of argument by developing a new and more powerful kind of staticity theorem 18,20 for "non-rotating" black holes: instead of

147

the litigious assumption of a lower bound on V (which was needed in the now obsolete Hawking-Lichnerowicz approach) the new theorem depends on the justifiable 21 requirement that there exists a slicing by a maximal (spacelike) hypersurface. It has thereby been possible to provide 19,25,84 a much more more satisfactorily complete demonstration of what had been rather overconfidently asserted by Hawking and Ellis 9, namely that subject to the assumptions of analyticity and of connectedness and non-degeneracy (c :f:. 0) of the horizon, the black hole equilibrium state has to be axisymmetric or static. In so far as most of the other essential steps referred to above are concerned, introductory presentations of the key technical details are already available elsewhere in surveys such as my 1987 review 17 and the very extensive and up to date treatise that has recently been provided by Heusler 26. However these surveys do not include any description of the technicalities of the new improved variety of staticity theorem 18,20, whose original presentation 18,20 was somewhat obscured by extraneous complications introduced in the (so far unfulfilled) hope of generalising the result to include Yang Mills fields. As an appendix to the present survey, I have therefore provided a brief but self contained account of the way this new kind of staticity theorem is obtained in the simplest case - namely that of a pure (Einstein) vacuum. 5

What remains for the future?

As Chrusciel has emphasised 84, although many of the (declared and hidden) assumptions involved in the work during the "classical phase" have been disposed of, the more recent work on the black hole equilibrium problem is still subject to several important technical restrictions whose treatment remains as a challenge for the future. One whose treatment should I think be given priority at this stage is the assumption of analyticity that has been invoked in all the work on the indispensible "strong rigidity" theorem that is needed to establish axisymmetry. It is to be remarked that if, as in my early work 5, axisymmetry is simply postulated at the outset, then analyticity will be demonstrable as an automatic consequence of the ellipticity of tM differential system that is obtained as a result of the "circularity" property that is established by the generalised Papapetrou theorem 80,5. What I would guess is that it should be possible (and probably not more difficult than the other steps that have already been acheived) to prove the necessity of analyticity for a vacuum equilibrium state it without assuming axisymmetry. A more delicate question that remains to be settled is the possibility of equilibrium involving several disconnected black holes. As far as the pure vacuum problem is concerned, my conjecture is that such multi black hole solutions do not exist, but they have so far been rigorously excluded only in the strictly static case 83. The axisymmetric case has recently been studied in some detail 85,86 by Weinstein (who denotes the horizon scale parameter c = ",A/47r by the letter JL) but a definitive conclusion has not yet emerged. In the electrovac case the situation is certainly more complicated, since it is known 51 that there are counterexamples in which gravitational attraction is balanced by electrostatic repulsion. It is however to be noticed

148

that the only counterexamples discovered so far, namely those of the PapapetrouMajumdar family 53,54 have horizons that are degenerate (in the sense of having a vanishing decay constant 1\;). It seems reasonable to conjecture that even in the electrovac case there are no non-degenerate multi black hole equilibrium states. This last point leads on to a third major problem that still needs to be dealt with, namely the general treatment of the degenerate (I\; = 0) case. The maximally rotating (J2 = M4) Kerr solution is still the only known pure vacuum example, and I am still inclined to conjecture that it is unique, but the problem of proving this remains entirely unsolved. As far as the electrovac problem is concerned, the only known examples are those of the Kerr-Newman 52 and Papapetrou-Majumdar 53 ,54 families. Recent progress by Heusler 88 has confirmed that the latter (whose equilibrium saturates a Bogomolny type mass limit 89 ) are the only strictly static examples, but for the rotating degenerate case the problem remains wide open. I wish to conclude by drawing attention to another deeper problem that, unlike the three referred to above, has been largely overlooked even by the experts in the field, but that seems to me just as interesting from a purely mathematical point of view, even if its physical relevance is less evident. This fourth problem, is that of solving the black hole equilibrium state problem without invoking the causality axiom on which nearly all the work described above depends (e.g. for obtaining the required positivity in the successive divergence identities 5,10,11,13,15,16,14 used in the axisymmetric case). As remarked in the introduction, all non static Kerr solutions contain closed timelike lines, though in the black hole subfamily with J2 :::; M4 they are entirely confined inside the horizon 2,27. Unlike analyticity, whose failure in shock type phenomena is physically familiar in many contexts, causality meaning the absence of closed timelike lines - is a requirement that most physicists would be prepared to take for granted as an indispensible requirement for realism in any classical field model. However the example of sphalerons suggests that despite their unacceptability at a classical level, the mathematical existence of stationary black hole states with closed timelike lines outside the horizon might have physically relevant implications in quantum theory. The discovery of such exotic configurations would be a surprise to most of us, but would not contradict any theorem obtained so far. All that can be confidently asserted at this stage is that such configurations could not be static but would have to be of rotating type. Appendix: the new staticity theorem

In view of the importance of the new kind of staticity theorem developed 18,20 by Sudarsky and Wald (superceding the Lichnerowicz kind, whose adaptation to the black hole context was inadequate in the pure vacuum case dealt with by Hawking 9 , and even less satisfactory in the electromagnetic case dealt by myself 8,17), this appendix presents a brief but self contained summary of the essential ideas. The Sudarsky Wald approach works perfectly well for the electrovac case (though it does seem to have trouble with Yang Mills fields) but in order to display the key points in the simplest possible form the description below is limited to the case of a pure (Einstein) vacuum. It is first to be recalled that the vacuum black hole equilibrium configurations

149

under consideration belong to the more general category of equilibrium configurations, including those of isolated states of self gravitating bodies such as neutron star models, that are invariant under the action of a time translation group whose generator kJ1. is timelike at least at large sufficiently large distance in the asymptotically flat outer region - where it can be taken to. be normalised so that its magnitude tends to unity at large distance. Any such configuration will of course have a well defined assymptotic mass, M say, given by a formula of the standard Komar form (1)

(using a semi-colon to indicate covariant differentiation) where the integral is taken over a surrounding topologically spherical spacelike 2-surface Soo whose choice is arbitrarily adjustable without affecting the result provided it is taken sufficiently far out to be entirely in the vacuum region where the source free Einstein equations are satisfied. In the pure vacuum black hole case with which we are concerned here, the analogous integral defining the black hole mass contribution

(2) in terms of any spacelike 2-surface S1-£ on the horizon will give the same result: the vanishing of the Riemann tensor RJ1.v in conjunction with the Killing equation

(3) (using round brackets for ansymmetrisation) ensures that divergence condition = 0 is satisfied all the way in to the horizon, with the implication that that M1-£ = M. According to Hawking's "strong rigidity theorem" 9 (which depends on the not yet satisfactorily justified analyticity postulate 25 dicussed above) the null tangent vector of the horizon will be normalisable in such a way as to coincide with a "corotating" Killing vector field IP given by a formula of the standard form kJ1.;V;v

(4) where n1-£ is a uniform ("rigid") angular velocity and hI' is indeterminate if n1-£ = 0 (i.e. in the "non-rotating" case) and otherwise is a well defined (axisymmetry Killing vector - with circular trajectories such that the correspondingly normalised angle parameter has period 27r. (If, instead of assuming analyticity, one assumes the existence of the axisymmetry generated by hI' then the formula (4) is very easily derivable by my "weak rigidity" theorm 66). The scale constant c of the horizon is defined by an expression of the same form as that for the mass but with the original (asymptotically timelike) Killing vector kJ1. replaced by the new the Killing vector combination IP that is null on the horizon, i.e. it is specified by _ 1 { OJ1.;V dS c - 47r } 1-£ {. J1.V

,

(5)

150

In terms of the acceleration parameter K given by 2K2 = 1/-l;Vlv;/-I which (like {}1£) must be uniform over the horizon by the "zeroth" law 69, the scale constant works out locally 8 as

KA

(6)

C=-,

411" where A is the horizon area. By the "rigidity" formula (4) (whether obtained from the "weak theorem" 66 assuming axisymmetry, or from the "strong theorem" 9 assuming analyticity) it immediately follows that the scale parameter will be expressible in terms of globally defined quantities via the Smarr type relation

(7) where M1£ is the black hole mass contribution as defined above and J1/. is the corresponding black hole angular momentum contribution,

J1/. = -

1~1I"

£

h/-l;v d)/-IV ,

(8)

which will be the same as the total angular momentum J -- - _1_ 1611"

1

J=

h/-l;v..Je' LU/-IV,

(9)

in the pure vacuum case considered here, i.e. we shall have J1/. = J. The new idea in the relatively recent work of Wald and Sudarsky 18,20 is to compare the long well known formula (7) that has just been recapitulated, which in the pure vacuum case under consideration here is evidently equivalent to the simple global mass formula (10) with a mass formula of the Arnowitt Deser Misner kind, which involves integration over a spacelike 3-surface ~ say. The trick used by Wald and Sudarsky was to choose the hypersurface ~ to be maximal - a restriction that has been confirmed to be imposable without loss of generality by Wald and Chrusciel 19 . This means that its second fundamental form, as expressed (using Latin letters for internal coordinate indices on the hypersurface) by Kij should be trace free, i.e. Ki i = 0 (using the induced 3-metric. i.e. the first fundamental form, for index raising). In the vacuum case this reduces the A.D.M. formula to the simple form

(11) where the surface field A is given in terms of the unit normal n/-l to ~ by A = -k/-ln/-l (which will be positive). Since the axisymmetry generator h/-l will be tangential to a section ~ that is maximal, one will have h/-ln/-l = 0, so that the "rigidity" formula (4) will simply give A = -l/-ln/-l on the horizon. The boundary 2-surface contribution from the horizon can thus be evaluated as

(12)

151

where c is the scale parameter as defined by (5). Identifying the output of this A.D.M. type mass formula (12) with that of our older Smarr type formula (10), one obtains a relationship expressible (in the pure vacuum case under consideration here) by

M - c = 20'H.J =

~ 47r

r Kij Kij>'d~.

iE

(13)

The manifest non-negativity of the integrand on the right hand side of the Wald Sudarsky identity (13) evidently entails that in the non rotating case the second fundamental form must vanish, i.e.

=>

(14)

Having thus established the extrinsic flatness of the maximal hypersurface for the case of a stationary black hole with non-rotating horizon, one can straightforwardly proceed to show that as a consequence the vector tJl. defined by (15) will automaticall satisfy a Killing equation t(JI.;v) = 0 of the same form as the one (3) satisfied by the original time translation generator, with which it will therefore be identifiable, i.e. one obtains (16) Since tJl. is hypersurface orthogonal by its construction (15), the desired staticity theorem is thereby established: it has been shown that the vanishing of the black hole angular velocity O'H. is sufficient by itself (without the need to postulate a questionable lower limit on the magnitude V = -kJl.kJl. as in the older treatment) to ensure hypersurface orthogonality of the time translation symmetry generator kJl.. References 1. B. Carter, "Stationary Axisymmetric Systems in General Relativity" (Ph.D. Thesis, DAMTP, Cambridge, 1977). 2. "Global Structure of the Kerr Family of Gravitational Fields", B. Carter, Phys. Rev. 174,1559-71 (1968). 3. R.P. Kerr, "Gravitational field of a spinning mass as an example of algebraically special metrics", Phys. Rev. Lett. 11, 237-38 (1963). 4. W. Israel, "Event horizons in static vacuum spacetimes", Phys.Rev 164, 177679 (1967). 5. B. Carter, "An Axisymmetric Black Hole has only Two Degrees of Freedom", B. Carter, Phys. Rev. Letters 26, 331-33 (1971). 6. S.W. Hawking "Black holes in General Relativity", Commun. Math. Phys. 25, 152-56 (1972). 7. S.W. Hawking, "The event horizon", in it Black Holes, (proc. 1972 Les Houches Summer School), ed. B. & C. DeWitt, 1-55 (Gordon and Breach, New York, 1973).

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8. B. Carter, "Black Hole Equilibrium States: II General Theory of Stationary Black Hole States", in Black Holes (proc. 1972 Les Houches Summer School), ed. B. & C. DeWitt, 125-210 (Gordon and Breach, New York, 1973). 9. S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space Time, (Cambridge U.P., 1973). 10. D.C. Robinson, "Classification of black holes with electromagnetic fields", Phys, Rev. 010, 458-60 (1974) 11. D.C. Robinson, "Uniqueness of the Kerr black hole", Phys, Rev. Lett.34, 905-06 (1975) 12. D.C. Robinson, "A simple proof of the generalisation of Israel's theorem", Gen. Rel. Grav. 8,695-98 (1977). 13. G. Bunting, "Proof of the Uniqueness Conjecture for Black Holes,"(Ph.D. Thesis, University of New England, Armadale N.S.W., 1983). 14. B. Carter, "The Bunting Identity and Mazur Identity for non-linear Elliptic Systems including the Black Hole Equilibrium Problem", Commun. Math. Phys. 99, 563-91 (1985). 15. P.O. Mazur, "Proof of uniqueness of the Kerr-Newman black hole solution", J. Phys. A15, 3173-80 (1982). 16. P.O. Mazur, "Black hole uniqueness from a hidden symmetry of Einstein's gravity", Gen. Rei Grav. 16, 211-15 (1984). 17. B. Carter "Mathematical foundations of the theory of relativistic stellar and black hole configurations", in Gravitation in Astrophysics (NATO ASI B156, Cargese, 1986), ed. B. Carter, J.B. Hartle, 63-122 (Plenum Press, New York, 1987). 18. D. Sudarsky, R.M. Wald, "Extrema of mass, stationarity, and staticity, and solutions to the Einstein-Yang-Mills equations", Phys. Rev. 046, 1453-74 (1991) 19. P.T Chrusciel, R.M. Wald, "Maximal hypersurfaces in stationary asymptotically flat spacetimes", Comm. Math. Phys. 163, 561-604 (1964). [grqc/9304009j 20. D. Sudarsky, R.M. Wald, "Mass formulas for stationary Einstein-Yang-Mills black holes and a simple proof of two staticity theorems", Phys. Rev. 047, 5209-13 (1993). [gr-qc/9305023j 21. P. T. Chrusciel, R.M. Wald, "On the topology of stationary black holes" , Class. Quantum Grav. 11, L147-52 (1994). [gr-qc/9410004j 22. K.S. Thorne, "Black holes and time warps" (Norton, New York, 1994). 23. W. Israel, "Dark stars: the evolution of an idea", in 300 years of gravitation, ed. S.W. Hawking, W. Israel, 199-276 (1987). 24. S.W. Hawking, "Particle creation by black holes", Comm. Math. Phys 43, 199-220 (1975). 25. P.T. Chrusciel, "No-hair theorems: folkelore, conjectures, results", in Differential Geometry and Mathematical Physics 170, ed J.Beem, K.L. Dugal, 23-49 (American Math. Soc., Providence, 1994) [gr-qc/9402032j 26. M. Heusler, Black hole uniqueness theorems (Cambridge U.P., 1996) 27. B.Carter, "Domains of stationary communications", Gen. Rel. Grav. 11, 437-50 (1978).

153

28. "Uber das Gravitationsfeld eines Massenpunctes nach der Einsteinschen Theorie" Sitzber. Deut. Akad. Wiss. Berlin KI. Math-Phys. Tech., 189-96 (1916) 29. J. Eisenstaedt, "Histoire et singularites de la solution de Schwarzschild (19151923)", Arch. Hist. Exact Sci 27, 157-228 (1982). 30. G.D. Birkhoff, Relativity and Modern Physics (Harvard U.P., 1923). 31. J.R Oppenheimer, H. Snyder "On continued gravitational contraction" , Phys. Rev. 56, 455-59 (1939). 32. P. Painleve, "La mecanique classique et la theorie de la relativite", C.R. Acad. Sci. (Paris) 173, 677-80 (1921). 33. A. Gullstrand, "Allegemeinne Losung des statischen Einkorper-problems in der Einsteinschen Gravitations theorie", Archiv. Mat. Astron. Fys. 16(8), 1-15 (1922). 34. A.S. Eddington, "A comparison of Whitehead's and Einstein's formulas", Nature 113, 192 (1924). 35. G. Lemaitre, "L'univers en expansion", Ann. Soc. Sci. Bruxelles I A53, 51-85 (1933). 36. J.L. Synge "The gravitational field of a particle", Proc. R. Irish Acad. A53, 83-114 (1950). 37. D. Finkelstein "Past-future asymmetry of a point particle", Phys. Rev. 110 965-67 (1958). 38. C. Frondsal "Completion and embedding of the Schwarzschild solution" , Phys. Rev. 116, 778-81 (1959). 39. M.D. Kruskal, "Minimal extension of the Schwarzschild metric", Phys. Rev. 119,1743-45 (1960). 40. G. Szekeres, "On the singularities of a Riemannian manifold", Publ. Math. Debrecen 7, 285-301 (1960) 41. Ya. B. Zel'dovich, LD. Novikov, Relativistic Astrophysics (Izdatel'stvo "Nauka", Moscow 1967); English version ed. KS Thorne, W.D. Arnett (University of Chicag